{"id":1448,"date":"2023-06-05T14:51:50","date_gmt":"2023-06-05T14:51:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/continuity\/"},"modified":"2023-06-05T14:51:50","modified_gmt":"2023-06-05T14:51:50","slug":"continuity","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/continuity\/","title":{"raw":"Continuity","rendered":"Continuity"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li style=\"font-weight: 400;\">Determine whether a function is continuous at a number.<\/li>\n \t<li style=\"font-weight: 400;\">Determine the input values for which a function is discontinuous.<\/li>\n<\/ul>\n<\/div>\nLet\u2019s consider a specific example of temperature in terms of date and location, such as June 27, 2013, in Phoenix, AZ. The graph in&nbsp;Figure 1&nbsp;indicates that, at 2 a.m., the temperature was [latex]{96}^{\\circ }\\text{F}[\/latex] . By 2 p.m. the temperature had risen to [latex]{116}^{\\circ }\\text{F,}[\/latex] and by 4 p.m. it was [latex]{118}^{\\circ }\\text{F}\\text{.}[\/latex] Sometime between 2 a.m. and 4 p.m., the temperature outside must have been exactly [latex]{110.5}^{\\circ }\\text{F}\\text{.}[\/latex] In fact, any temperature between [latex]{96}^{\\circ }\\text{F}[\/latex] and [latex]{118}^{\\circ }\\text{F}[\/latex] occurred at some point that day. This means all real numbers in the output between [latex]{96}^{\\circ }\\text{F}[\/latex] and [latex]{118}^{\\circ }\\text{F}[\/latex] are generated at some point by the function according to the intermediate value theorem,\n\nLook again at Figure 1. There are no breaks in the function\u2019s graph for this 24-hour period. At no point did the temperature cease to exist, nor was there a point at which the temperature jumped instantaneously by several degrees. A function that has no holes or breaks in its graph is known as a <strong>continuous function<\/strong>. Temperature as a function of time is an example of a continuous function.\n\nIf temperature represents a continuous function, what kind of function would not be continuous? Consider an example of dollars expressed as a function of hours of parking. Let\u2019s create the function [latex]D[\/latex], where [latex]D\\left(x\\right)[\/latex] is the output representing cost in dollars for parking [latex]x[\/latex] number of hours.\n\nSuppose a parking garage charges $4.00 per hour or fraction of an hour, with a $24 per day maximum charge. Park for two hours and five minutes and the charge is $12. Park an additional hour and the charge is $16. We can never be charged $13, $14, or $15. There are real numbers between 12 and 16 that the function never outputs. There are breaks in the function\u2019s graph for this 24-hour period, points at which the price of parking jumps instantaneously by several dollars.\n\n[caption id=\"\" align=\"aligncenter\" width=\"977\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185310\/CNX_Precalc_Figure_12_03_0022.jpg\" alt=\"Graph of function that maps the time since midnight to the temperature. The x-axis represents the hours parked from 0 to 24. The y-axis represents dollars amounting from 0 to 28. The function is a step-function.\" width=\"977\" height=\"361\"> <b>Figure 2.<\/b> Parking-garage charges form a discontinuous function.[\/caption]\n\nA function that remains level for an interval and then jumps instantaneously to a higher value is called a <strong>stepwise function<\/strong>. This function is an example.\n\nA function that has any hole or break in its graph is known as a <strong>discontinuous function<\/strong>. A stepwise function, such as parking-garage charges as a function of hours parked, is an example of a discontinuous function.\n\nSo how can we decide if a function is continuous at a particular number? We can check three different conditions. Let\u2019s use the function [latex]y=f\\left(x\\right)[\/latex] represented in Figure 3&nbsp;as an example.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185313\/CNX_Precalc_Figure_12_03_0032.jpg\" alt=\"Graph of an increasing function with a discontinuity at (a, f(a)).\" width=\"487\" height=\"251\"> <b>Figure 3<\/b>[\/caption]\n\n<strong>Condition 1<\/strong> According to Condition 1, the function [latex]f\\left(a\\right)[\/latex] defined at [latex]x=a[\/latex] must exist. In other words, there is a <em>y<\/em>-coordinate at [latex]x=a[\/latex] as in Figure 4.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185315\/CNX_Precalc_Figure_12_03_0042.jpg\" alt=\"Graph of an increasing function with a discontinuity at (a, 2). The point (a, f(a)) is directly below the hole.\" width=\"487\" height=\"251\"> <b>Figure 4<\/b>[\/caption]\n\n<strong>Condition 2<\/strong> According to Condition 2, at [latex]x=a[\/latex] the limit, written [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)[\/latex], must exist. This means that at [latex]x=a[\/latex] the left-hand limit must equal the right-hand limit. Notice as the graph of [latex]f[\/latex] in Figure 3&nbsp;approaches [latex]x=a[\/latex] from the left and right, the same <em>y<\/em>-coordinate is approached. Therefore, Condition 2 is satisfied. However, there could still be a hole in the graph at [latex]x=a[\/latex] .\n\n<strong>Condition 3<\/strong> According to Condition 3, the corresponding [latex]y[\/latex] coordinate at [latex]x=a[\/latex] fills in the hole in the graph of [latex]f[\/latex]. This is written [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=f\\left(a\\right)[\/latex].\n\nSatisfying all three conditions means that the function is continuous. All three conditions are satisfied for the function represented in Figure 5&nbsp;so the function is continuous as [latex]x=a[\/latex].\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185319\/CNX_Precalc_Figure_12_03_0052.jpg\" alt=\"Graph of an increasing function with filled-in discontinuity at (a, f(a)).\" width=\"487\" height=\"251\"> <b>Figure 5.<\/b> All three conditions are satisfied. The function is continuous at [latex]x=a[\/latex] .[\/caption]Figure 6 through Figure 9&nbsp;provide several examples of graphs of functions that are not continuous at [latex]x=a[\/latex] and the condition or conditions that fail.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185322\/CNX_Precalc_Figure_12_03_0062.jpg\" alt=\"Graph of an increasing function with a discontinuity at (a, f(a)).\" width=\"487\" height=\"251\"> <b>Figure 6.<\/b> Condition 2 is satisfied. Conditions 1 and 3 both fail.[\/caption]\n\n&nbsp;\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185324\/CNX_Precalc_Figure_12_03_0072.jpg\" alt=\"Graph of an increasing function with a discontinuity at (a, 2). The point (a, f(a)) is directly below the hole.\" width=\"487\" height=\"256\"> <b>Figure 7.<\/b> Conditions 1 and 2 are both satisfied. Condition 3 fails.[\/caption]\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185327\/CNX_Precalc_Figure_12_03_0082.jpg\" alt=\"Graph of a piecewise function with an increasing segment from negative infinity to (a, f(a)), which is closed, and another increasing segment from (a, f(a)-1), which is open, to positive infinity.\" width=\"487\" height=\"250\"> <b>Figure 8.<\/b> Condition 1 is satisfied. Conditions 2 and 3 fail.[\/caption]\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185329\/CNX_Precalc_Figure_12_03_0092.jpg\" alt=\"Graph of a piecewise function with an increasing segment from negative infinity to (a, f(a)) and another increasing segment from (a, f(a) - 1) to positive infinity. This graph does not include the point (a, f(a)).\" width=\"487\" height=\"251\"> <b>Figure 9.<\/b> Conditions 1, 2, and 3 all fail.[\/caption]\n\n<div class=\"textbox\">\n<h3>A General Note: Definition of Continuity<\/h3>\nA function [latex]f\\left(x\\right)[\/latex] is <strong>continuous<\/strong> at [latex]x=a[\/latex] provided all three of the following conditions hold true:\n\nCondition 1: [latex]f(a)[\/latex] exists.\n\nCondition 2: [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)[\/latex] exists at [latex]x=a[\/latex].\n\nCondition 2: [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)=f(a)[\/latex].\n\nIf a function [latex]f\\left(x\\right)[\/latex] is not continuous at [latex]x=a[\/latex], the function is <strong>discontinuous<\/strong> at [latex]x=a[\/latex] .\n\n<\/div>\n<h2>Identifying Discontinuities<\/h2>\nDiscontinuity can occur in different ways. We saw in the previous section that a function could have a <strong>left-hand limit<\/strong> and a <strong>right-hand limit<\/strong> even if they are not equal. If the left- and right-hand limits exist but are different, the graph \"jumps\" at [latex]x=a[\/latex] . The function is said to have a jump discontinuity.\n\nAs an example, look at the graph of the function [latex]y=f\\left(x\\right)[\/latex] in Figure 10. Notice as [latex]x[\/latex] approaches [latex]a[\/latex] how the output approaches different values from the left and from the right.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185331\/CNX_Precalc_Figure_12_03_0102.jpg\" alt=\"Graph of a piecewise function with an increasing segment from negative infinity to (a, f(a)), which is closed, and another increasing segment from (a, f(a)-1), which is open, to positive infinity.\" width=\"487\" height=\"251\"> <b>Figure 10.<\/b> Graph of a function with a jump discontinuity.[\/caption]\n\n<div class=\"textbox\">\n<h3>&nbsp;A General Note: Jump Discontinuity<\/h3>\nA function [latex]f\\left(x\\right)[\/latex] has a <strong>jump discontinuity<\/strong> at [latex]x=a[\/latex] if the left- and right-hand limits both exist but are not equal: [latex]\\underset{x\\to {a}^{-}}{\\mathrm{lim}}f\\left(x\\right)\\ne \\underset{x\\to {a}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex] .\n\n<\/div>\n<h2>Identifying Removable Discontinuity<\/h2>\nSome functions have a discontinuity, but it is possible to redefine the function at that point to make it continuous. This type of function is said to have a removable discontinuity. Let\u2019s look at the function [latex]y=f\\left(x\\right)[\/latex] represented by the graph in Figure 11. The function has a limit. However, there is a hole at [latex]x=a[\/latex] . The hole can be filled by extending the domain to include the input [latex]x=a[\/latex] and defining the corresponding output of the function at that value as the limit of the function at [latex]x=a[\/latex] .\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185333\/CNX_Precalc_Figure_12_03_0112.jpg\" alt=\"Graph of an increasing function with a removable discontinuity at (a, f(a)).\" width=\"487\" height=\"251\"> <b>Figure 11.<\/b> Graph of function [latex]f[\/latex] with a removable discontinuity at [latex]x=a[\/latex] .[\/caption]\n<div class=\"textbox\">\n<h3>A General Note: Removable Discontinuity<\/h3>\nA function [latex]f\\left(x\\right)[\/latex] has a <strong>removable discontinuity<\/strong> at [latex]x=a[\/latex] if the limit, [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)[\/latex], exists, but either\n<div>\n<ol>\n \t<li>[latex]f\\left(a\\right)[\/latex] does not exist <em>or<\/em><\/li>\n \t<li>[latex]f\\left(a\\right)[\/latex], the value of the function at [latex]x=a[\/latex] does not equal the limit, [latex]f\\left(a\\right)\\ne \\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 1: Identifying Discontinuities<\/h3>\nIdentify all discontinuities for the following functions as either a jump or a removable discontinuity.\n<ol>\n \t<li>[latex]f\\left(x\\right)=\\frac{{x}^{2}-2x - 15}{x - 5}[\/latex]<\/li>\n \t<li>[latex]g\\left(x\\right)=\\begin{cases}x+1, \\hfill&amp; x&lt;2 \\\\ -x, \\hfill&amp; x\\geq2\\end{cases}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"277675\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"277675\"]\n<ol>\n \t<li>Notice that the function is defined everywhere except at [latex]x=5[\/latex].Thus, [latex]f\\left(5\\right)[\/latex] does not exist, Condition 2 is not satisfied. Since Condition 1 is satisfied, the limit as [latex]x[\/latex] approaches 5 is 8, and Condition 2 is not satisfied.This means there is a removable discontinuity at [latex]x=5[\/latex].<\/li>\n \t<li>Condition 2 is satisfied because [latex]g\\left(2\\right)=-2[\/latex].Notice that the function is a <strong>piecewise function<\/strong>, and for each piece, the function is defined everywhere on its domain. Let\u2019s examine Condition 1 by determining the left- and right-hand limits as [latex]x[\/latex] approaches 2.Left-hand limit: [latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}\\left(x+1\\right)=2+1=3[\/latex]. The left-hand limit exists.Right-hand limit: [latex]\\underset{x\\to {2}^{+}}{\\mathrm{lim}}\\left(-x\\right)=-2[\/latex]. The right-hand limit exists. But\n<div style=\"text-align: center;\">[latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)\\ne \\underset{x\\to {2}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex].<\/div>\nSo, [latex]\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)[\/latex] does not exist, and Condition 2 fails: There is no removable discontinuity. However, since both left- and right-hand limits exist but are not equal, the conditions are satisfied for a jump discontinuity at [latex]x=2[\/latex].<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nIdentify all discontinuities for the following functions as either a jump or a removable discontinuity.\n<p style=\"padding-left: 60px;\">a. [latex]f\\left(x\\right)=\\frac{{x}^{2}-6x}{x - 6}[\/latex]<\/p>\n<p style=\"padding-left: 60px;\">b. [latex]g\\left(x\\right)=\\begin{cases}\\sqrt{x}, \\hfill&amp; 0\\leq x&lt;4 \\\\ 2x, \\hfill&amp; x\\geq4\\end{cases}[\/latex]<\/p>\n[reveal-answer q=\"284168\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"284168\"]\n\na.&nbsp;removable discontinuity at [latex]x=6[\/latex];\nb. jump discontinuity at [latex]x=4[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\n<h2>Recognizing Continuous and Discontinuous Real-Number Functions<\/h2>\nMany of the functions we have encountered in earlier chapters are continuous everywhere. They never have a hole in them, and they never jump from one value to the next. For all of these functions, the limit of [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] is the same as the value of [latex]f\\left(x\\right)[\/latex] when [latex]x=a[\/latex]. So [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=f\\left(a\\right)[\/latex]. There are some functions that are continuous everywhere and some that are only continuous where they are defined on their domain because they are not defined for all real numbers.\n<div class=\"textbox\">\n<h3>A General Note: Examples of Continuous Functions<\/h3>\nThe following functions are continuous everywhere:\n<table><colgroup> <col> <col> <\/colgroup>\n<tbody>\n<tr>\n<td style=\"width: 122.386px;\">Polynomial functions<\/td>\n<td style=\"width: 469.659px;\">Ex: [latex]f\\left(x\\right)={x}^{4}-9{x}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 122.386px;\">Exponential functions<\/td>\n<td style=\"width: 469.659px;\">Ex: [latex]f\\left(x\\right)={4}^{x+2}-5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 122.386px;\">Sine functions<\/td>\n<td style=\"width: 469.659px;\">Ex: [latex]f\\left(x\\right)=\\sin \\left(2x\\right)-4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 122.386px;\">Cosine functions<\/td>\n<td style=\"width: 469.659px;\">Ex: [latex]f\\left(x\\right)=-\\cos \\left(x+\\frac{\\pi }{3}\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nThe following functions are continuous everywhere they are defined on their domain:\n<table><colgroup> <col> <col> <\/colgroup>\n<tbody>\n<tr>\n<td>Logarithmic functions<\/td>\n<td>Ex: [latex]f\\left(x\\right)=2\\mathrm{ln}\\left(x\\right)[\/latex] , [latex]x&gt;0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Tangent functions<\/td>\n<td>Ex: [latex]f\\left(x\\right)=\\tan \\left(x\\right)+2[\/latex], [latex]x\\ne \\frac{\\pi }{2}+k\\pi [\/latex], [latex]k[\/latex] is an integer<\/td>\n<\/tr>\n<tr>\n<td>Rational functions<\/td>\n<td>Ex: [latex]f\\left(x\\right)=\\frac{{x}^{2}-25}{x - 7}[\/latex], [latex]x\\ne 7[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function [latex]\\begin{align}f\\left(x\\right)\\end{align}[\/latex], determine if the function is continuous at [latex]\\begin{align}x=a\\end{align}[\/latex].<\/h3>\n<ol>\n \t<li>Check Condition 1: [latex]f\\left(a\\right)[\/latex] exists.<\/li>\n \t<li>Check Condition 2: [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)[\/latex] exists at [latex]x=a[\/latex].<\/li>\n \t<li>Check Condition 3: [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=f\\left(a\\right)[\/latex].<\/li>\n \t<li>If all three conditions are satisfied, the function is continuous at [latex]x=a[\/latex]. If any one of the conditions is not satisfied, the function is not continuous at [latex]x=a[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 2: Determining Whether a Piecewise Function is Continuous at a Given Number<\/h3>\nDetermine whether the function [latex]f\\left(x\\right)=\\begin{cases}4x, \\hfill&amp; x\\leq 3 \\\\ 8+x, \\hfill&amp; x&gt;3\\end{cases}[\/latex] is continuous at\n<ol>\n \t<li>[latex]x=3[\/latex]<\/li>\n \t<li>[latex]x=\\frac{8}{3}[\/latex]<\/li>\n<\/ol>\n[reveal-answer q=\"324975\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"324975\"]\n\nTo determine if the function [latex]f[\/latex] is continuous at [latex]x=a[\/latex], we will determine if the three conditions of continuity are satisfied at [latex]x=a[\/latex] .\n<ol>\n \t<li>Condition 1: Does [latex]f\\left(a\\right)[\/latex] exist?\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;f\\left(3\\right)=4\\left(3\\right)=12 \\\\ &amp;\\Rightarrow \\text{Condition 1 is satisfied}. \\end{align}[\/latex]<\/div>\nCondition 2: Does [latex]\\underset{x\\to 3}{\\mathrm{lim}}f\\left(x\\right)[\/latex] exist?\n\nTo the left of [latex]x=3[\/latex], [latex]f\\left(x\\right)=4x[\/latex]; to the right of [latex]x=3[\/latex], [latex]f\\left(x\\right)=8+x[\/latex]. We need to evaluate the left- and right-hand limits as [latex]x[\/latex] approaches 1.\n<div style=\"margin: 0 0 0 40px; border: none; padding: 0px;\">Left-hand limit: [latex]\\underset{x\\to {3}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {3}^{-}}{\\mathrm{lim}}4\\left(3\\right)=12[\/latex]\nRight-hand limit: [latex]\\underset{x\\to {3}^{+}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {3}^{+}}{\\mathrm{lim}}\\left(8+x\\right)=8+3=11[\/latex]<\/div>\n<span style=\"font-size: 1rem; text-align: initial;\">Because [latex]\\underset{x\\to {1}^{-}}{\\mathrm{lim}}f\\left(x\\right)\\ne \\underset{x\\to {1}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex], [latex]\\underset{x\\to 1}{\\mathrm{lim}}f\\left(x\\right)[\/latex] does not exist.\n<\/span>\n<div style=\"text-align: center;\">[latex]\\Rightarrow \\text{ Condition 2 fails}\\text{.}[\/latex]<\/div>\nThere is no need to proceed further. Condition 2 fails at [latex]x=3[\/latex]. If any of the conditions of continuity are not satisfied at [latex]x=3[\/latex], the function [latex]f\\left(x\\right)[\/latex] is not continuous at [latex]x=3[\/latex].<\/li>\n \t<li>[latex]x=\\frac{8}{3}[\/latex]Condition 1: Does [latex]f\\left(\\frac{8}{3}\\right)[\/latex] exist?\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;f\\left(\\frac{8}{3}\\right)=4\\left(\\frac{8}{3}\\right)=\\frac{32}{3} \\\\ &amp;\\Rightarrow \\text{Condition 1 is satisfied}. \\end{align}[\/latex]<\/div>\n<span style=\"font-size: 1rem; text-align: initial;\">Condition 2: Does [latex]\\underset{x\\to \\frac{8}{3}}{\\mathrm{lim}}f\\left(x\\right)[\/latex] exist?<\/span>\n\nTo the left of [latex]x=\\frac{8}{3}[\/latex], [latex]f\\left(x\\right)=4x[\/latex]; to the right of [latex]x=\\frac{8}{3}[\/latex], [latex]f\\left(x\\right)=8+x[\/latex]. We need to evaluate the left- and right-hand limits as [latex]x[\/latex] approaches [latex]\\frac{8}{3}[\/latex].\n<div>\n<div>\n<div style=\"margin: 0 0 0 40px; border: none; padding: 0px;\">Left-hand limit: [latex]\\underset{x\\to {\\frac{8}{3}}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {\\frac{8}{3}}^{-}}{\\mathrm{lim}}4\\left(\\frac{8}{3}\\right)=\\frac{32}{3}[\/latex]\nRight-hand limit: [latex]\\underset{x\\to {\\frac{8}{3}}^{+}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {\\frac{8}{3}}^{+}}{\\mathrm{lim}}\\left(8+x\\right)=8+\\frac{8}{3}=\\frac{32}{3}[\/latex]<\/div>\n<\/div>\n<\/div>\nBecause [latex]\\underset{x\\to \\frac{8}{3}}{\\mathrm{lim}}f\\left(x\\right)[\/latex] exists,\n<div style=\"text-align: center;\">[latex]\\Rightarrow \\text{Condition 2 is satisfied}[\/latex].<\/div>\nCondition 3: Is [latex]f\\left(\\frac{8}{3}\\right)=\\underset{x\\to \\frac{8}{3}}{\\mathrm{lim}}f\\left(x\\right)?[\/latex]\n<div style=\"text-align: center;\">[latex]\\begin{align}f&amp;\\left(\\frac{32}{3}\\right)=\\frac{32}{3}=\\underset{x\\to \\frac{8}{3}}{\\mathrm{lim}}f\\left(x\\right) \\\\ &amp;\\Rightarrow \\text{Condition 3 is satisfied}. \\end{align}[\/latex]<\/div>\nBecause all three conditions of continuity are satisfied at [latex]x=\\frac{8}{3}[\/latex], the function [latex]f\\left(x\\right)[\/latex] is continuous at [latex]x=\\frac{8}{3}[\/latex].\n<ol>\n \t<li style=\"list-style-type: none;\">[\/hidden-answer]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nDetermine whether the function [latex]f\\left(x\\right)=\\begin{cases}\\frac{1}{x}, \\hfill&amp; x\\leq 2 \\\\ 9x-17.5, \\hfill&amp; x&gt;2\\end{cases}[\/latex] is continuous at [latex]x=2[\/latex].\n\n[reveal-answer q=\"968995\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"968995\"]\n\nyes\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Determining Whether a Rational Function is Continuous at a Given Number<\/h3>\nDetermine whether the function [latex]f\\left(x\\right)=\\frac{{x}^{2}-25}{x - 5}[\/latex] is continuous at [latex]x=5[\/latex].\n\n[reveal-answer q=\"48479\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"48479\"]\n\nTo determine if the function [latex]f[\/latex] is continuous at [latex]x=5[\/latex], we will determine if the three conditions of continuity are satisfied at [latex]x=5[\/latex].\n\nCondition 1:\n<p style=\"text-align: center;\">[latex]\\begin{gathered}f\\left(5\\right)\\text{ does not exist.} \\\\ \\Rightarrow \\text{Condition 1 fails}.\\end{gathered}[\/latex]<\/p>\nThere is no need to proceed further. Condition 2 fails at [latex]x=5[\/latex]. If any of the conditions of continuity are not satisfied at [latex]x=5[\/latex], the function [latex]f[\/latex] is not continuous at [latex]x=5[\/latex].\n<h4>Analysis of the Solution<\/h4>\nSee Figure 12. Notice that for Condition 2 we have\n<p style=\"text-align: center;\">[latex]\\begin{align}\\underset{x\\to 5}{\\mathrm{lim}}\\frac{{x}^{2}-25}{x - 5}&amp;=\\underset{x\\to 3}{\\mathrm{lim}}\\frac{\\cancel{\\left(x - 5\\right)}\\left(x+5\\right)}{\\cancel{x - 5}} \\\\ &amp;=\\underset{x\\to 5}{\\mathrm{lim}}\\left(x+5\\right) \\\\ &amp;=5+5=10 \\\\ \\Rightarrow \\text{Conditio}&amp;\\text{n 2 is satisfied}. \\end{align}[\/latex]<\/p>\nAt [latex]x=5[\/latex], there exists a removable discontinuity.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185335\/CNX_Precalc_Figure_12_03_0132.jpg\" alt=\"Graph of an increasing function with a removable discontinuity at (5, 10).\" width=\"487\" height=\"570\"> <b>Figure 12<\/b>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nDetermine whether the function [latex]f\\left(x\\right)=\\frac{9-{x}^{2}}{{x}^{2}-3x}[\/latex] is continuous at [latex]x=3[\/latex]. If not, state the type of discontinuity.\n\n[reveal-answer q=\"252333\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"252333\"]\n\nNo, the function is not continuous at [latex]x=3[\/latex]. There exists a removable discontinuity at [latex]x=3[\/latex].\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question hide_question_numbers=1]174101[\/ohm_question]\n\n<\/div>\n<h2>Determining the Input Values for Which a Function Is Discontinuous<\/h2>\nNow that we can identify continuous functions, jump discontinuities, and removable discontinuities, we will look at more complex functions to find discontinuities. Here, we will analyze a piecewise function to determine if any real numbers exist where the function is not continuous. A <strong>piecewise function<\/strong> may have discontinuities at the boundary points of the function as well as within the functions that make it up.\n\nTo determine the real numbers for which a piecewise function composed of polynomial functions is not continuous, recall that <strong>polynomial<\/strong> functions themselves are continuous on the set of real numbers. Any discontinuity would be at the boundary points. So we need to explore the three conditions of continuity at the boundary points of the piecewise function.\n<div class=\"textbox\">\n<h3>How To: Given a piecewise function, determine whether it is continuous at the boundary points.<\/h3>\n<div>\n<ol>\n \t<li>For each boundary point [latex]a[\/latex] of the piecewise function, determine the left- and right-hand limits as [latex]x[\/latex] approaches [latex]a[\/latex], as well as the function value at [latex]a[\/latex].<\/li>\n \t<li>Check each condition for each value to determine if all three conditions are satisfied.<\/li>\n \t<li>Determine whether each value satisfies condition 1: [latex]f\\left(a\\right)[\/latex] exists.<\/li>\n \t<li>Determine whether each value satisfies condition 2: [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)[\/latex] exists.<\/li>\n \t<li>Determine whether each value satisfies condition 3: [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=f\\left(a\\right)[\/latex].<\/li>\n \t<li>If all three conditions are satisfied, the function is continuous at [latex]x=a[\/latex]. If any one of the conditions fails, the function is not continuous at [latex]x=a[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 4: Determining the Input Values for Which a Piecewise Function Is Discontinuous<\/h3>\nDetermine whether the function [latex]f[\/latex] is discontinuous for any real numbers.\n<p style=\"text-align: center;\">[latex]fx=\\begin{cases}x+1, \\hfill&amp; x&lt;2 \\\\ 3, \\hfill&amp; 2\\leq x&lt;4 \\\\ x^{2}-11, \\hfill&amp; x\\geq 4\\end{cases}[\/latex]<\/p>\n[reveal-answer q=\"28596\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"28596\"]\n\nThe piecewise function is defined by three functions, which are all polynomial functions, [latex]f\\left(x\\right)=x+1[\/latex] on [latex]x&lt;2[\/latex], [latex]f\\left(x\\right)=3[\/latex] on [latex]2\\le x&lt;4[\/latex], and [latex]f\\left(x\\right)={x}^{2}-5[\/latex] on [latex]x\\ge 4[\/latex]. Polynomial functions are continuous everywhere. Any discontinuities would be at the boundary points, [latex]x=2[\/latex] and [latex]x=4[\/latex].\n\nAt [latex]x=2[\/latex], let us check the three conditions of continuity.\n\nCondition 1:\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;f\\left(2\\right)=3\\\\ &amp;\\Rightarrow \\text{Condition 1 is satisfied}.\\end{align}[\/latex]<\/p>\nCondition 2: Because a different function defines the output left and right of [latex]x=2[\/latex], does [latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {2}^{+}}{\\mathrm{lim}}f\\left(x\\right)?[\/latex]\n<div>\n<ul>\n \t<li>Left-hand limit: [latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {2}^{-}}{\\mathrm{lim}}\\left(x+1\\right)=2+1=3[\/latex]<\/li>\n \t<li>Right-hand limit: [latex]\\underset{x\\to {2}^{+}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {2}^{+}}{\\mathrm{lim}}3=3[\/latex]<\/li>\n<\/ul>\n<\/div>\nBecause [latex]3=3[\/latex] , [latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {2}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]\n<p style=\"text-align: center;\">[latex]\\Rightarrow \\text{Condition 2 is satisfied}[\/latex].<\/p>\nCondition 3:\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)=3=f\\left(2\\right)\\\\ &amp;\\Rightarrow \\text{Condition 3 is satisfied}.\\end{align}[\/latex]<\/p>\nBecause all three conditions are satisfied at [latex]x=2[\/latex], the function [latex]f\\left(x\\right)[\/latex] is continuous at [latex]x=2[\/latex].\n\nAt [latex]x=4[\/latex], let us check the three conditions of continuity.\n\nCondition 2: Because a different function defines the output left and right of [latex]x=4[\/latex], does [latex]\\underset{x\\to {4}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {4}^{+}}{\\mathrm{lim}}f\\left(x\\right)?[\/latex]\n<div>\n<ul>\n \t<li>Left-hand limit: [latex]\\underset{x\\to {4}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {4}^{-}}{\\mathrm{lim}}3=3[\/latex]<\/li>\n \t<li>Right-hand limit: [latex]\\underset{x\\to {4}^{+}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {4}^{+}}{\\mathrm{lim}}\\left({x}^{2}-11\\right)={4}^{2}-11=5[\/latex]<\/li>\n<\/ul>\n<\/div>\nBecause [latex]3\\ne 5[\/latex] , [latex]\\underset{x\\to {4}^{-}}{\\mathrm{lim}}f\\left(x\\right)\\ne \\underset{x\\to {4}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex], so [latex]\\underset{x\\to 4}{\\mathrm{lim}}f\\left(x\\right)[\/latex] does not exist.\n<p style=\"text-align: center;\">[latex]\\Rightarrow \\text{Condition 2 fails}[\/latex].<\/p>\nBecause one of the three conditions does not hold at [latex]x=4[\/latex], the function [latex]f\\left(x\\right)[\/latex] is discontinuous at [latex]x=4[\/latex].\n<h4>Analysis of the Solution<\/h4>\nAt [latex]x=4[\/latex], there exists a jump discontinuity. Notice that the function is continuous at [latex]x=2[\/latex].\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185337\/CNX_Precalc_Figure_12_03_0142.jpg\" alt=\"Graph of a piecewise function that has disconuity at (4, 3).\" width=\"487\" height=\"476\"> <b>Figure 13.<\/b> Graph is continuous at [latex]x=2[\/latex] but shows a jump discontinuity at [latex]x=4[\/latex].[\/caption][\/hidden-answer]<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\nDetermine where the function\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\begin{cases}\\frac{\\pi x}{4}, &amp;x&lt;2 \\\\ \\frac{\\pi}{x}, &amp;2\\leq x \\leq 6 \\\\ 2\\pi x, &amp;x&gt;6\\end{cases}[\/latex]<\/p>\nis discontinuous.\n\n[reveal-answer q=\"358632\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"358632\"]\n\n[latex]x=6[\/latex]\n\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n[ohm_question hide_question_numbers=1]174104[\/ohm_question]\n\n<\/div>\n<h2>Determining Whether a Function Is Continuous<\/h2>\nTo determine whether a <strong>piecewise function<\/strong> is continuous or discontinuous, in addition to checking the boundary points, we must also check whether each of the functions that make up the piecewise function is continuous.\n<div class=\"textbox\">\n<h3>How To: Given a piecewise function, determine whether it is continuous.<\/h3>\n<ol>\n \t<li>Determine whether each component function of the piecewise function is continuous. If there are discontinuities, do they occur within the domain where that component function is applied?<\/li>\n \t<li>For each boundary point [latex]x=a[\/latex] of the piecewise function, determine if each of the three conditions hold.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 5: Determining Whether a Piecewise Function Is Continuous<\/h3>\nDetermine whether the function below is continuous. If it is not, state the location and type of each discontinuity.\n<p style=\"text-align: center;\">[latex]fx=\\begin{cases}sin\\left(x\\right), \\hfill&amp; x&lt;0 \\\\ x^{3}, \\hfill&amp; x&gt;0\\end{cases}[\/latex]<\/p>\n[reveal-answer q=\"177808\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"177808\"]\n\nThe two functions composing this piecewise function are [latex]f\\left(x\\right)=\\sin \\left(x\\right)[\/latex] on [latex]x&lt;0[\/latex] and [latex]f\\left(x\\right)={x}^{3}[\/latex] on [latex]x&gt;0[\/latex]. The sine function and all polynomial functions are continuous everywhere. Any discontinuities would be at the boundary point,\n\nAt [latex]x=0[\/latex], let us check the three conditions of continuity.\n\nCondition 1:\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;f\\left(0\\right)\\text{ does not exist}. \\\\ &amp;\\Rightarrow \\text{Condition 1 fails}. \\end{align}[\/latex]<\/p>\nBecause all three conditions are not satisfied at [latex]x=0[\/latex], the function [latex]f\\left(x\\right)[\/latex] is discontinuous at [latex]x=0[\/latex].\n<h4>Analysis of the Solution<\/h4>\nThere exists a removable discontinuity at [latex]x=0[\/latex]; [latex]\\underset{x\\to 0}{\\mathrm{lim}}f\\left(x\\right)=0[\/latex], thus the limit exists and is finite, but [latex]f\\left(a\\right)[\/latex] does not exist.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185339\/CNX_Precalc_Figure_12_03_0152.jpg\" alt=\"Graph of a piecewise function where from negative infinity to 0 f(x) = sin(x) and from 0 to positive infinity f(x) = x^3.\" width=\"487\" height=\"370\"> <b>Figure 14.<\/b> Function has removable discontinuity at 0.[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>A continuous function can be represented by a graph without holes or breaks.<\/li>\n \t<li>A function whose graph has holes is a discontinuous function.<\/li>\n \t<li>A function is continuous at a particular number if three conditions are met:\n<ul>\n \t<li>Condition 1: [latex]f\\left(a\\right)[\/latex] exists.<\/li>\n \t<li>Condition 2: [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)[\/latex] exists at [latex]x=a[\/latex].<\/li>\n \t<li>Condition 3: [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=f\\left(a\\right)[\/latex].<\/li>\n<\/ul>\n<\/li>\n \t<li>A function has a jump discontinuity if the left- and right-hand limits are different, causing the graph to \"jump.\"<\/li>\n \t<li>A function has a removable discontinuity if it can be redefined at its discontinuous point to make it continuous.<\/li>\n \t<li>Some functions, such as polynomial functions, are continuous everywhere. Other functions, such as logarithmic functions, are continuous on their domain.<\/li>\n \t<li>For a piecewise function to be continuous each piece must be continuous on its part of the domain and the function as a whole must be continuous at the boundaries.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135699897\" class=\"definition\">\n \t<dt>continuous function<\/dt>\n \t<dd id=\"fs-id1165135699903\">a function that has no holes or breaks in its graph<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135699907\" class=\"definition\">\n \t<dt>discontinuous function<\/dt>\n \t<dd id=\"fs-id1165135699912\">a function that is not continuous at [latex]x=a[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135699930\" class=\"definition\">\n \t<dt>jump discontinuity<\/dt>\n \t<dd id=\"fs-id1165135699935\">a point of discontinuity in a function [latex]f\\left(x\\right)[\/latex] at [latex]x=a[\/latex] where both the left and right-hand limits exist, but [latex]\\underset{x\\to {a}^{-}}{\\mathrm{lim}}f\\left(x\\right)\\ne \\underset{x\\to {a}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135700059\" class=\"definition\">\n \t<dt>removable discontinuity<\/dt>\n \t<dd id=\"fs-id1165135700065\">a point of discontinuity in a function [latex]f\\left(x\\right)[\/latex] where the function is discontinuous, but can be redefined to make it continuous<\/dd>\n<\/dl>\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li style=\"font-weight: 400;\">Determine whether a function is continuous at a number.<\/li>\n<li style=\"font-weight: 400;\">Determine the input values for which a function is discontinuous.<\/li>\n<\/ul>\n<\/div>\n<p>Let\u2019s consider a specific example of temperature in terms of date and location, such as June 27, 2013, in Phoenix, AZ. The graph in&nbsp;Figure 1&nbsp;indicates that, at 2 a.m., the temperature was [latex]{96}^{\\circ }\\text{F}[\/latex] . By 2 p.m. the temperature had risen to [latex]{116}^{\\circ }\\text{F,}[\/latex] and by 4 p.m. it was [latex]{118}^{\\circ }\\text{F}\\text{.}[\/latex] Sometime between 2 a.m. and 4 p.m., the temperature outside must have been exactly [latex]{110.5}^{\\circ }\\text{F}\\text{.}[\/latex] In fact, any temperature between [latex]{96}^{\\circ }\\text{F}[\/latex] and [latex]{118}^{\\circ }\\text{F}[\/latex] occurred at some point that day. This means all real numbers in the output between [latex]{96}^{\\circ }\\text{F}[\/latex] and [latex]{118}^{\\circ }\\text{F}[\/latex] are generated at some point by the function according to the intermediate value theorem,<\/p>\n<p>Look again at Figure 1. There are no breaks in the function\u2019s graph for this 24-hour period. At no point did the temperature cease to exist, nor was there a point at which the temperature jumped instantaneously by several degrees. A function that has no holes or breaks in its graph is known as a <strong>continuous function<\/strong>. Temperature as a function of time is an example of a continuous function.<\/p>\n<p>If temperature represents a continuous function, what kind of function would not be continuous? Consider an example of dollars expressed as a function of hours of parking. Let\u2019s create the function [latex]D[\/latex], where [latex]D\\left(x\\right)[\/latex] is the output representing cost in dollars for parking [latex]x[\/latex] number of hours.<\/p>\n<p>Suppose a parking garage charges $4.00 per hour or fraction of an hour, with a $24 per day maximum charge. Park for two hours and five minutes and the charge is $12. Park an additional hour and the charge is $16. We can never be charged $13, $14, or $15. There are real numbers between 12 and 16 that the function never outputs. There are breaks in the function\u2019s graph for this 24-hour period, points at which the price of parking jumps instantaneously by several dollars.<\/p>\n<div style=\"width: 987px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185310\/CNX_Precalc_Figure_12_03_0022.jpg\" alt=\"Graph of function that maps the time since midnight to the temperature. The x-axis represents the hours parked from 0 to 24. The y-axis represents dollars amounting from 0 to 28. The function is a step-function.\" width=\"977\" height=\"361\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2.<\/b> Parking-garage charges form a discontinuous function.<\/p>\n<\/div>\n<p>A function that remains level for an interval and then jumps instantaneously to a higher value is called a <strong>stepwise function<\/strong>. This function is an example.<\/p>\n<p>A function that has any hole or break in its graph is known as a <strong>discontinuous function<\/strong>. A stepwise function, such as parking-garage charges as a function of hours parked, is an example of a discontinuous function.<\/p>\n<p>So how can we decide if a function is continuous at a particular number? We can check three different conditions. Let\u2019s use the function [latex]y=f\\left(x\\right)[\/latex] represented in Figure 3&nbsp;as an example.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185313\/CNX_Precalc_Figure_12_03_0032.jpg\" alt=\"Graph of an increasing function with a discontinuity at (a, f(a)).\" width=\"487\" height=\"251\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\n<p><strong>Condition 1<\/strong> According to Condition 1, the function [latex]f\\left(a\\right)[\/latex] defined at [latex]x=a[\/latex] must exist. In other words, there is a <em>y<\/em>-coordinate at [latex]x=a[\/latex] as in Figure 4.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185315\/CNX_Precalc_Figure_12_03_0042.jpg\" alt=\"Graph of an increasing function with a discontinuity at (a, 2). The point (a, f(a)) is directly below the hole.\" width=\"487\" height=\"251\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 4<\/b><\/p>\n<\/div>\n<p><strong>Condition 2<\/strong> According to Condition 2, at [latex]x=a[\/latex] the limit, written [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)[\/latex], must exist. This means that at [latex]x=a[\/latex] the left-hand limit must equal the right-hand limit. Notice as the graph of [latex]f[\/latex] in Figure 3&nbsp;approaches [latex]x=a[\/latex] from the left and right, the same <em>y<\/em>-coordinate is approached. Therefore, Condition 2 is satisfied. However, there could still be a hole in the graph at [latex]x=a[\/latex] .<\/p>\n<p><strong>Condition 3<\/strong> According to Condition 3, the corresponding [latex]y[\/latex] coordinate at [latex]x=a[\/latex] fills in the hole in the graph of [latex]f[\/latex]. This is written [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=f\\left(a\\right)[\/latex].<\/p>\n<p>Satisfying all three conditions means that the function is continuous. All three conditions are satisfied for the function represented in Figure 5&nbsp;so the function is continuous as [latex]x=a[\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185319\/CNX_Precalc_Figure_12_03_0052.jpg\" alt=\"Graph of an increasing function with filled-in discontinuity at (a, f(a)).\" width=\"487\" height=\"251\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 5.<\/b> All three conditions are satisfied. The function is continuous at [latex]x=a[\/latex] .<\/p>\n<\/div>\n<p>Figure 6 through Figure 9&nbsp;provide several examples of graphs of functions that are not continuous at [latex]x=a[\/latex] and the condition or conditions that fail.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185322\/CNX_Precalc_Figure_12_03_0062.jpg\" alt=\"Graph of an increasing function with a discontinuity at (a, f(a)).\" width=\"487\" height=\"251\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6.<\/b> Condition 2 is satisfied. Conditions 1 and 3 both fail.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185324\/CNX_Precalc_Figure_12_03_0072.jpg\" alt=\"Graph of an increasing function with a discontinuity at (a, 2). The point (a, f(a)) is directly below the hole.\" width=\"487\" height=\"256\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7.<\/b> Conditions 1 and 2 are both satisfied. Condition 3 fails.<\/p>\n<\/div>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185327\/CNX_Precalc_Figure_12_03_0082.jpg\" alt=\"Graph of a piecewise function with an increasing segment from negative infinity to (a, f(a)), which is closed, and another increasing segment from (a, f(a)-1), which is open, to positive infinity.\" width=\"487\" height=\"250\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 8.<\/b> Condition 1 is satisfied. Conditions 2 and 3 fail.<\/p>\n<\/div>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185329\/CNX_Precalc_Figure_12_03_0092.jpg\" alt=\"Graph of a piecewise function with an increasing segment from negative infinity to (a, f(a)) and another increasing segment from (a, f(a) - 1) to positive infinity. This graph does not include the point (a, f(a)).\" width=\"487\" height=\"251\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 9.<\/b> Conditions 1, 2, and 3 all fail.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Definition of Continuity<\/h3>\n<p>A function [latex]f\\left(x\\right)[\/latex] is <strong>continuous<\/strong> at [latex]x=a[\/latex] provided all three of the following conditions hold true:<\/p>\n<p>Condition 1: [latex]f(a)[\/latex] exists.<\/p>\n<p>Condition 2: [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)[\/latex] exists at [latex]x=a[\/latex].<\/p>\n<p>Condition 2: [latex]\\underset{x\\to a}{\\mathrm{lim}}f(x)=f(a)[\/latex].<\/p>\n<p>If a function [latex]f\\left(x\\right)[\/latex] is not continuous at [latex]x=a[\/latex], the function is <strong>discontinuous<\/strong> at [latex]x=a[\/latex] .<\/p>\n<\/div>\n<h2>Identifying Discontinuities<\/h2>\n<p>Discontinuity can occur in different ways. We saw in the previous section that a function could have a <strong>left-hand limit<\/strong> and a <strong>right-hand limit<\/strong> even if they are not equal. If the left- and right-hand limits exist but are different, the graph &#8220;jumps&#8221; at [latex]x=a[\/latex] . The function is said to have a jump discontinuity.<\/p>\n<p>As an example, look at the graph of the function [latex]y=f\\left(x\\right)[\/latex] in Figure 10. Notice as [latex]x[\/latex] approaches [latex]a[\/latex] how the output approaches different values from the left and from the right.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185331\/CNX_Precalc_Figure_12_03_0102.jpg\" alt=\"Graph of a piecewise function with an increasing segment from negative infinity to (a, f(a)), which is closed, and another increasing segment from (a, f(a)-1), which is open, to positive infinity.\" width=\"487\" height=\"251\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 10.<\/b> Graph of a function with a jump discontinuity.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>&nbsp;A General Note: Jump Discontinuity<\/h3>\n<p>A function [latex]f\\left(x\\right)[\/latex] has a <strong>jump discontinuity<\/strong> at [latex]x=a[\/latex] if the left- and right-hand limits both exist but are not equal: [latex]\\underset{x\\to {a}^{-}}{\\mathrm{lim}}f\\left(x\\right)\\ne \\underset{x\\to {a}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex] .<\/p>\n<\/div>\n<h2>Identifying Removable Discontinuity<\/h2>\n<p>Some functions have a discontinuity, but it is possible to redefine the function at that point to make it continuous. This type of function is said to have a removable discontinuity. Let\u2019s look at the function [latex]y=f\\left(x\\right)[\/latex] represented by the graph in Figure 11. The function has a limit. However, there is a hole at [latex]x=a[\/latex] . The hole can be filled by extending the domain to include the input [latex]x=a[\/latex] and defining the corresponding output of the function at that value as the limit of the function at [latex]x=a[\/latex] .<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185333\/CNX_Precalc_Figure_12_03_0112.jpg\" alt=\"Graph of an increasing function with a removable discontinuity at (a, f(a)).\" width=\"487\" height=\"251\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 11.<\/b> Graph of function [latex]f[\/latex] with a removable discontinuity at [latex]x=a[\/latex] .<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Removable Discontinuity<\/h3>\n<p>A function [latex]f\\left(x\\right)[\/latex] has a <strong>removable discontinuity<\/strong> at [latex]x=a[\/latex] if the limit, [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)[\/latex], exists, but either<\/p>\n<div>\n<ol>\n<li>[latex]f\\left(a\\right)[\/latex] does not exist <em>or<\/em><\/li>\n<li>[latex]f\\left(a\\right)[\/latex], the value of the function at [latex]x=a[\/latex] does not equal the limit, [latex]f\\left(a\\right)\\ne \\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 1: Identifying Discontinuities<\/h3>\n<p>Identify all discontinuities for the following functions as either a jump or a removable discontinuity.<\/p>\n<ol>\n<li>[latex]f\\left(x\\right)=\\frac{{x}^{2}-2x - 15}{x - 5}[\/latex]<\/li>\n<li>[latex]g\\left(x\\right)=\\begin{cases}x+1, \\hfill& x<2 \\\\ -x, \\hfill& x\\geq2\\end{cases}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q277675\">Show Solution<\/span><\/p>\n<div id=\"q277675\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Notice that the function is defined everywhere except at [latex]x=5[\/latex].Thus, [latex]f\\left(5\\right)[\/latex] does not exist, Condition 2 is not satisfied. Since Condition 1 is satisfied, the limit as [latex]x[\/latex] approaches 5 is 8, and Condition 2 is not satisfied.This means there is a removable discontinuity at [latex]x=5[\/latex].<\/li>\n<li>Condition 2 is satisfied because [latex]g\\left(2\\right)=-2[\/latex].Notice that the function is a <strong>piecewise function<\/strong>, and for each piece, the function is defined everywhere on its domain. Let\u2019s examine Condition 1 by determining the left- and right-hand limits as [latex]x[\/latex] approaches 2.Left-hand limit: [latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}\\left(x+1\\right)=2+1=3[\/latex]. The left-hand limit exists.Right-hand limit: [latex]\\underset{x\\to {2}^{+}}{\\mathrm{lim}}\\left(-x\\right)=-2[\/latex]. The right-hand limit exists. But\n<div style=\"text-align: center;\">[latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)\\ne \\underset{x\\to {2}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex].<\/div>\n<p>So, [latex]\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)[\/latex] does not exist, and Condition 2 fails: There is no removable discontinuity. However, since both left- and right-hand limits exist but are not equal, the conditions are satisfied for a jump discontinuity at [latex]x=2[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Identify all discontinuities for the following functions as either a jump or a removable discontinuity.<\/p>\n<p style=\"padding-left: 60px;\">a. [latex]f\\left(x\\right)=\\frac{{x}^{2}-6x}{x - 6}[\/latex]<\/p>\n<p style=\"padding-left: 60px;\">b. [latex]g\\left(x\\right)=\\begin{cases}\\sqrt{x}, \\hfill& 0\\leq x<4 \\\\ 2x, \\hfill& x\\geq4\\end{cases}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q284168\">Show Solution<\/span><\/p>\n<div id=\"q284168\" class=\"hidden-answer\" style=\"display: none\">\n<p>a.&nbsp;removable discontinuity at [latex]x=6[\/latex];<br \/>\nb. jump discontinuity at [latex]x=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Recognizing Continuous and Discontinuous Real-Number Functions<\/h2>\n<p>Many of the functions we have encountered in earlier chapters are continuous everywhere. They never have a hole in them, and they never jump from one value to the next. For all of these functions, the limit of [latex]f\\left(x\\right)[\/latex] as [latex]x[\/latex] approaches [latex]a[\/latex] is the same as the value of [latex]f\\left(x\\right)[\/latex] when [latex]x=a[\/latex]. So [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=f\\left(a\\right)[\/latex]. There are some functions that are continuous everywhere and some that are only continuous where they are defined on their domain because they are not defined for all real numbers.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Examples of Continuous Functions<\/h3>\n<p>The following functions are continuous everywhere:<\/p>\n<table>\n<colgroup>\n<col \/>\n<col \/> <\/colgroup>\n<tbody>\n<tr>\n<td style=\"width: 122.386px;\">Polynomial functions<\/td>\n<td style=\"width: 469.659px;\">Ex: [latex]f\\left(x\\right)={x}^{4}-9{x}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 122.386px;\">Exponential functions<\/td>\n<td style=\"width: 469.659px;\">Ex: [latex]f\\left(x\\right)={4}^{x+2}-5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 122.386px;\">Sine functions<\/td>\n<td style=\"width: 469.659px;\">Ex: [latex]f\\left(x\\right)=\\sin \\left(2x\\right)-4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 122.386px;\">Cosine functions<\/td>\n<td style=\"width: 469.659px;\">Ex: [latex]f\\left(x\\right)=-\\cos \\left(x+\\frac{\\pi }{3}\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The following functions are continuous everywhere they are defined on their domain:<\/p>\n<table>\n<colgroup>\n<col \/>\n<col \/> <\/colgroup>\n<tbody>\n<tr>\n<td>Logarithmic functions<\/td>\n<td>Ex: [latex]f\\left(x\\right)=2\\mathrm{ln}\\left(x\\right)[\/latex] , [latex]x>0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Tangent functions<\/td>\n<td>Ex: [latex]f\\left(x\\right)=\\tan \\left(x\\right)+2[\/latex], [latex]x\\ne \\frac{\\pi }{2}+k\\pi[\/latex], [latex]k[\/latex] is an integer<\/td>\n<\/tr>\n<tr>\n<td>Rational functions<\/td>\n<td>Ex: [latex]f\\left(x\\right)=\\frac{{x}^{2}-25}{x - 7}[\/latex], [latex]x\\ne 7[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function [latex]\\begin{align}f\\left(x\\right)\\end{align}[\/latex], determine if the function is continuous at [latex]\\begin{align}x=a\\end{align}[\/latex].<\/h3>\n<ol>\n<li>Check Condition 1: [latex]f\\left(a\\right)[\/latex] exists.<\/li>\n<li>Check Condition 2: [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)[\/latex] exists at [latex]x=a[\/latex].<\/li>\n<li>Check Condition 3: [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=f\\left(a\\right)[\/latex].<\/li>\n<li>If all three conditions are satisfied, the function is continuous at [latex]x=a[\/latex]. If any one of the conditions is not satisfied, the function is not continuous at [latex]x=a[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 2: Determining Whether a Piecewise Function is Continuous at a Given Number<\/h3>\n<p>Determine whether the function [latex]f\\left(x\\right)=\\begin{cases}4x, \\hfill& x\\leq 3 \\\\ 8+x, \\hfill& x>3\\end{cases}[\/latex] is continuous at<\/p>\n<ol>\n<li>[latex]x=3[\/latex]<\/li>\n<li>[latex]x=\\frac{8}{3}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q324975\">Show Solution<\/span><\/p>\n<div id=\"q324975\" class=\"hidden-answer\" style=\"display: none\">\n<p>To determine if the function [latex]f[\/latex] is continuous at [latex]x=a[\/latex], we will determine if the three conditions of continuity are satisfied at [latex]x=a[\/latex] .<\/p>\n<ol>\n<li>Condition 1: Does [latex]f\\left(a\\right)[\/latex] exist?\n<div style=\"text-align: center;\">[latex]\\begin{align}&f\\left(3\\right)=4\\left(3\\right)=12 \\\\ &\\Rightarrow \\text{Condition 1 is satisfied}. \\end{align}[\/latex]<\/div>\n<p>Condition 2: Does [latex]\\underset{x\\to 3}{\\mathrm{lim}}f\\left(x\\right)[\/latex] exist?<\/p>\n<p>To the left of [latex]x=3[\/latex], [latex]f\\left(x\\right)=4x[\/latex]; to the right of [latex]x=3[\/latex], [latex]f\\left(x\\right)=8+x[\/latex]. We need to evaluate the left- and right-hand limits as [latex]x[\/latex] approaches 1.<\/p>\n<div style=\"margin: 0 0 0 40px; border: none; padding: 0px;\">Left-hand limit: [latex]\\underset{x\\to {3}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {3}^{-}}{\\mathrm{lim}}4\\left(3\\right)=12[\/latex]<br \/>\nRight-hand limit: [latex]\\underset{x\\to {3}^{+}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {3}^{+}}{\\mathrm{lim}}\\left(8+x\\right)=8+3=11[\/latex]<\/div>\n<p><span style=\"font-size: 1rem; text-align: initial;\">Because [latex]\\underset{x\\to {1}^{-}}{\\mathrm{lim}}f\\left(x\\right)\\ne \\underset{x\\to {1}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex], [latex]\\underset{x\\to 1}{\\mathrm{lim}}f\\left(x\\right)[\/latex] does not exist.<br \/>\n<\/span><\/p>\n<div style=\"text-align: center;\">[latex]\\Rightarrow \\text{ Condition 2 fails}\\text{.}[\/latex]<\/div>\n<p>There is no need to proceed further. Condition 2 fails at [latex]x=3[\/latex]. If any of the conditions of continuity are not satisfied at [latex]x=3[\/latex], the function [latex]f\\left(x\\right)[\/latex] is not continuous at [latex]x=3[\/latex].<\/li>\n<li>[latex]x=\\frac{8}{3}[\/latex]Condition 1: Does [latex]f\\left(\\frac{8}{3}\\right)[\/latex] exist?\n<div style=\"text-align: center;\">[latex]\\begin{align}&f\\left(\\frac{8}{3}\\right)=4\\left(\\frac{8}{3}\\right)=\\frac{32}{3} \\\\ &\\Rightarrow \\text{Condition 1 is satisfied}. \\end{align}[\/latex]<\/div>\n<p><span style=\"font-size: 1rem; text-align: initial;\">Condition 2: Does [latex]\\underset{x\\to \\frac{8}{3}}{\\mathrm{lim}}f\\left(x\\right)[\/latex] exist?<\/span><\/p>\n<p>To the left of [latex]x=\\frac{8}{3}[\/latex], [latex]f\\left(x\\right)=4x[\/latex]; to the right of [latex]x=\\frac{8}{3}[\/latex], [latex]f\\left(x\\right)=8+x[\/latex]. We need to evaluate the left- and right-hand limits as [latex]x[\/latex] approaches [latex]\\frac{8}{3}[\/latex].<\/p>\n<div>\n<div>\n<div style=\"margin: 0 0 0 40px; border: none; padding: 0px;\">Left-hand limit: [latex]\\underset{x\\to {\\frac{8}{3}}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {\\frac{8}{3}}^{-}}{\\mathrm{lim}}4\\left(\\frac{8}{3}\\right)=\\frac{32}{3}[\/latex]<br \/>\nRight-hand limit: [latex]\\underset{x\\to {\\frac{8}{3}}^{+}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {\\frac{8}{3}}^{+}}{\\mathrm{lim}}\\left(8+x\\right)=8+\\frac{8}{3}=\\frac{32}{3}[\/latex]<\/div>\n<\/div>\n<\/div>\n<p>Because [latex]\\underset{x\\to \\frac{8}{3}}{\\mathrm{lim}}f\\left(x\\right)[\/latex] exists,<\/p>\n<div style=\"text-align: center;\">[latex]\\Rightarrow \\text{Condition 2 is satisfied}[\/latex].<\/div>\n<p>Condition 3: Is [latex]f\\left(\\frac{8}{3}\\right)=\\underset{x\\to \\frac{8}{3}}{\\mathrm{lim}}f\\left(x\\right)?[\/latex]<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}f&\\left(\\frac{32}{3}\\right)=\\frac{32}{3}=\\underset{x\\to \\frac{8}{3}}{\\mathrm{lim}}f\\left(x\\right) \\\\ &\\Rightarrow \\text{Condition 3 is satisfied}. \\end{align}[\/latex]<\/div>\n<p>Because all three conditions of continuity are satisfied at [latex]x=\\frac{8}{3}[\/latex], the function [latex]f\\left(x\\right)[\/latex] is continuous at [latex]x=\\frac{8}{3}[\/latex].<\/p>\n<ol>\n<li style=\"list-style-type: none;\"><\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Determine whether the function [latex]f\\left(x\\right)=\\begin{cases}\\frac{1}{x}, \\hfill& x\\leq 2 \\\\ 9x-17.5, \\hfill& x>2\\end{cases}[\/latex] is continuous at [latex]x=2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q968995\">Show Solution<\/span><\/p>\n<div id=\"q968995\" class=\"hidden-answer\" style=\"display: none\">\n<p>yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Determining Whether a Rational Function is Continuous at a Given Number<\/h3>\n<p>Determine whether the function [latex]f\\left(x\\right)=\\frac{{x}^{2}-25}{x - 5}[\/latex] is continuous at [latex]x=5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q48479\">Show Solution<\/span><\/p>\n<div id=\"q48479\" class=\"hidden-answer\" style=\"display: none\">\n<p>To determine if the function [latex]f[\/latex] is continuous at [latex]x=5[\/latex], we will determine if the three conditions of continuity are satisfied at [latex]x=5[\/latex].<\/p>\n<p>Condition 1:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{gathered}f\\left(5\\right)\\text{ does not exist.} \\\\ \\Rightarrow \\text{Condition 1 fails}.\\end{gathered}[\/latex]<\/p>\n<p>There is no need to proceed further. Condition 2 fails at [latex]x=5[\/latex]. If any of the conditions of continuity are not satisfied at [latex]x=5[\/latex], the function [latex]f[\/latex] is not continuous at [latex]x=5[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>See Figure 12. Notice that for Condition 2 we have<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\underset{x\\to 5}{\\mathrm{lim}}\\frac{{x}^{2}-25}{x - 5}&=\\underset{x\\to 3}{\\mathrm{lim}}\\frac{\\cancel{\\left(x - 5\\right)}\\left(x+5\\right)}{\\cancel{x - 5}} \\\\ &=\\underset{x\\to 5}{\\mathrm{lim}}\\left(x+5\\right) \\\\ &=5+5=10 \\\\ \\Rightarrow \\text{Conditio}&\\text{n 2 is satisfied}. \\end{align}[\/latex]<\/p>\n<p>At [latex]x=5[\/latex], there exists a removable discontinuity.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185335\/CNX_Precalc_Figure_12_03_0132.jpg\" alt=\"Graph of an increasing function with a removable discontinuity at (5, 10).\" width=\"487\" height=\"570\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 12<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Determine whether the function [latex]f\\left(x\\right)=\\frac{9-{x}^{2}}{{x}^{2}-3x}[\/latex] is continuous at [latex]x=3[\/latex]. If not, state the type of discontinuity.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q252333\">Show Solution<\/span><\/p>\n<div id=\"q252333\" class=\"hidden-answer\" style=\"display: none\">\n<p>No, the function is not continuous at [latex]x=3[\/latex]. There exists a removable discontinuity at [latex]x=3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174101\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174101&theme=oea&iframe_resize_id=ohm174101\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Determining the Input Values for Which a Function Is Discontinuous<\/h2>\n<p>Now that we can identify continuous functions, jump discontinuities, and removable discontinuities, we will look at more complex functions to find discontinuities. Here, we will analyze a piecewise function to determine if any real numbers exist where the function is not continuous. A <strong>piecewise function<\/strong> may have discontinuities at the boundary points of the function as well as within the functions that make it up.<\/p>\n<p>To determine the real numbers for which a piecewise function composed of polynomial functions is not continuous, recall that <strong>polynomial<\/strong> functions themselves are continuous on the set of real numbers. Any discontinuity would be at the boundary points. So we need to explore the three conditions of continuity at the boundary points of the piecewise function.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a piecewise function, determine whether it is continuous at the boundary points.<\/h3>\n<div>\n<ol>\n<li>For each boundary point [latex]a[\/latex] of the piecewise function, determine the left- and right-hand limits as [latex]x[\/latex] approaches [latex]a[\/latex], as well as the function value at [latex]a[\/latex].<\/li>\n<li>Check each condition for each value to determine if all three conditions are satisfied.<\/li>\n<li>Determine whether each value satisfies condition 1: [latex]f\\left(a\\right)[\/latex] exists.<\/li>\n<li>Determine whether each value satisfies condition 2: [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)[\/latex] exists.<\/li>\n<li>Determine whether each value satisfies condition 3: [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=f\\left(a\\right)[\/latex].<\/li>\n<li>If all three conditions are satisfied, the function is continuous at [latex]x=a[\/latex]. If any one of the conditions fails, the function is not continuous at [latex]x=a[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 4: Determining the Input Values for Which a Piecewise Function Is Discontinuous<\/h3>\n<p>Determine whether the function [latex]f[\/latex] is discontinuous for any real numbers.<\/p>\n<p style=\"text-align: center;\">[latex]fx=\\begin{cases}x+1, \\hfill& x<2 \\\\ 3, \\hfill& 2\\leq x<4 \\\\ x^{2}-11, \\hfill& x\\geq 4\\end{cases}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q28596\">Show Solution<\/span><\/p>\n<div id=\"q28596\" class=\"hidden-answer\" style=\"display: none\">\n<p>The piecewise function is defined by three functions, which are all polynomial functions, [latex]f\\left(x\\right)=x+1[\/latex] on [latex]x<2[\/latex], [latex]f\\left(x\\right)=3[\/latex] on [latex]2\\le x<4[\/latex], and [latex]f\\left(x\\right)={x}^{2}-5[\/latex] on [latex]x\\ge 4[\/latex]. Polynomial functions are continuous everywhere. Any discontinuities would be at the boundary points, [latex]x=2[\/latex] and [latex]x=4[\/latex].\n\nAt [latex]x=2[\/latex], let us check the three conditions of continuity.\n\nCondition 1:\n\n\n<p style=\"text-align: center;\">[latex]\\begin{align}&f\\left(2\\right)=3\\\\ &\\Rightarrow \\text{Condition 1 is satisfied}.\\end{align}[\/latex]<\/p>\n<p>Condition 2: Because a different function defines the output left and right of [latex]x=2[\/latex], does [latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {2}^{+}}{\\mathrm{lim}}f\\left(x\\right)?[\/latex]<\/p>\n<div>\n<ul>\n<li>Left-hand limit: [latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {2}^{-}}{\\mathrm{lim}}\\left(x+1\\right)=2+1=3[\/latex]<\/li>\n<li>Right-hand limit: [latex]\\underset{x\\to {2}^{+}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {2}^{+}}{\\mathrm{lim}}3=3[\/latex]<\/li>\n<\/ul>\n<\/div>\n<p>Because [latex]3=3[\/latex] , [latex]\\underset{x\\to {2}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {2}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\Rightarrow \\text{Condition 2 is satisfied}[\/latex].<\/p>\n<p>Condition 3:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&\\underset{x\\to 2}{\\mathrm{lim}}f\\left(x\\right)=3=f\\left(2\\right)\\\\ &\\Rightarrow \\text{Condition 3 is satisfied}.\\end{align}[\/latex]<\/p>\n<p>Because all three conditions are satisfied at [latex]x=2[\/latex], the function [latex]f\\left(x\\right)[\/latex] is continuous at [latex]x=2[\/latex].<\/p>\n<p>At [latex]x=4[\/latex], let us check the three conditions of continuity.<\/p>\n<p>Condition 2: Because a different function defines the output left and right of [latex]x=4[\/latex], does [latex]\\underset{x\\to {4}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {4}^{+}}{\\mathrm{lim}}f\\left(x\\right)?[\/latex]<\/p>\n<div>\n<ul>\n<li>Left-hand limit: [latex]\\underset{x\\to {4}^{-}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {4}^{-}}{\\mathrm{lim}}3=3[\/latex]<\/li>\n<li>Right-hand limit: [latex]\\underset{x\\to {4}^{+}}{\\mathrm{lim}}f\\left(x\\right)=\\underset{x\\to {4}^{+}}{\\mathrm{lim}}\\left({x}^{2}-11\\right)={4}^{2}-11=5[\/latex]<\/li>\n<\/ul>\n<\/div>\n<p>Because [latex]3\\ne 5[\/latex] , [latex]\\underset{x\\to {4}^{-}}{\\mathrm{lim}}f\\left(x\\right)\\ne \\underset{x\\to {4}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex], so [latex]\\underset{x\\to 4}{\\mathrm{lim}}f\\left(x\\right)[\/latex] does not exist.<\/p>\n<p style=\"text-align: center;\">[latex]\\Rightarrow \\text{Condition 2 fails}[\/latex].<\/p>\n<p>Because one of the three conditions does not hold at [latex]x=4[\/latex], the function [latex]f\\left(x\\right)[\/latex] is discontinuous at [latex]x=4[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>At [latex]x=4[\/latex], there exists a jump discontinuity. Notice that the function is continuous at [latex]x=2[\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185337\/CNX_Precalc_Figure_12_03_0142.jpg\" alt=\"Graph of a piecewise function that has disconuity at (4, 3).\" width=\"487\" height=\"476\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 13.<\/b> Graph is continuous at [latex]x=2[\/latex] but shows a jump discontinuity at [latex]x=4[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Determine where the function<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=\\begin{cases}\\frac{\\pi x}{4}, &x<2 \\\\ \\frac{\\pi}{x}, &2\\leq x \\leq 6 \\\\ 2\\pi x, &x>6\\end{cases}[\/latex]<\/p>\n<p>is discontinuous.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q358632\">Show Solution<\/span><\/p>\n<div id=\"q358632\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x=6[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174104\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174104&theme=oea&iframe_resize_id=ohm174104\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Determining Whether a Function Is Continuous<\/h2>\n<p>To determine whether a <strong>piecewise function<\/strong> is continuous or discontinuous, in addition to checking the boundary points, we must also check whether each of the functions that make up the piecewise function is continuous.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a piecewise function, determine whether it is continuous.<\/h3>\n<ol>\n<li>Determine whether each component function of the piecewise function is continuous. If there are discontinuities, do they occur within the domain where that component function is applied?<\/li>\n<li>For each boundary point [latex]x=a[\/latex] of the piecewise function, determine if each of the three conditions hold.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 5: Determining Whether a Piecewise Function Is Continuous<\/h3>\n<p>Determine whether the function below is continuous. If it is not, state the location and type of each discontinuity.<\/p>\n<p style=\"text-align: center;\">[latex]fx=\\begin{cases}sin\\left(x\\right), \\hfill& x<0 \\\\ x^{3}, \\hfill& x>0\\end{cases}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q177808\">Show Solution<\/span><\/p>\n<div id=\"q177808\" class=\"hidden-answer\" style=\"display: none\">\n<p>The two functions composing this piecewise function are [latex]f\\left(x\\right)=\\sin \\left(x\\right)[\/latex] on [latex]x<0[\/latex] and [latex]f\\left(x\\right)={x}^{3}[\/latex] on [latex]x>0[\/latex]. The sine function and all polynomial functions are continuous everywhere. Any discontinuities would be at the boundary point,<\/p>\n<p>At [latex]x=0[\/latex], let us check the three conditions of continuity.<\/p>\n<p>Condition 1:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&f\\left(0\\right)\\text{ does not exist}. \\\\ &\\Rightarrow \\text{Condition 1 fails}. \\end{align}[\/latex]<\/p>\n<p>Because all three conditions are not satisfied at [latex]x=0[\/latex], the function [latex]f\\left(x\\right)[\/latex] is discontinuous at [latex]x=0[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>There exists a removable discontinuity at [latex]x=0[\/latex]; [latex]\\underset{x\\to 0}{\\mathrm{lim}}f\\left(x\\right)=0[\/latex], thus the limit exists and is finite, but [latex]f\\left(a\\right)[\/latex] does not exist.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27185339\/CNX_Precalc_Figure_12_03_0152.jpg\" alt=\"Graph of a piecewise function where from negative infinity to 0 f(x) = sin(x) and from 0 to positive infinity f(x) = x^3.\" width=\"487\" height=\"370\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 14.<\/b> Function has removable discontinuity at 0.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>A continuous function can be represented by a graph without holes or breaks.<\/li>\n<li>A function whose graph has holes is a discontinuous function.<\/li>\n<li>A function is continuous at a particular number if three conditions are met:\n<ul>\n<li>Condition 1: [latex]f\\left(a\\right)[\/latex] exists.<\/li>\n<li>Condition 2: [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)[\/latex] exists at [latex]x=a[\/latex].<\/li>\n<li>Condition 3: [latex]\\underset{x\\to a}{\\mathrm{lim}}f\\left(x\\right)=f\\left(a\\right)[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li>A function has a jump discontinuity if the left- and right-hand limits are different, causing the graph to &#8220;jump.&#8221;<\/li>\n<li>A function has a removable discontinuity if it can be redefined at its discontinuous point to make it continuous.<\/li>\n<li>Some functions, such as polynomial functions, are continuous everywhere. Other functions, such as logarithmic functions, are continuous on their domain.<\/li>\n<li>For a piecewise function to be continuous each piece must be continuous on its part of the domain and the function as a whole must be continuous at the boundaries.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135699897\" class=\"definition\">\n<dt>continuous function<\/dt>\n<dd id=\"fs-id1165135699903\">a function that has no holes or breaks in its graph<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135699907\" class=\"definition\">\n<dt>discontinuous function<\/dt>\n<dd id=\"fs-id1165135699912\">a function that is not continuous at [latex]x=a[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135699930\" class=\"definition\">\n<dt>jump discontinuity<\/dt>\n<dd id=\"fs-id1165135699935\">a point of discontinuity in a function [latex]f\\left(x\\right)[\/latex] at [latex]x=a[\/latex] where both the left and right-hand limits exist, but [latex]\\underset{x\\to {a}^{-}}{\\mathrm{lim}}f\\left(x\\right)\\ne \\underset{x\\to {a}^{+}}{\\mathrm{lim}}f\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135700059\" class=\"definition\">\n<dt>removable discontinuity<\/dt>\n<dd id=\"fs-id1165135700065\">a point of discontinuity in a function [latex]f\\left(x\\right)[\/latex] where the function is discontinuous, but can be redefined to make it continuous<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1448\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":503070,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1448","chapter","type-chapter","status-publish","hentry"],"part":1445,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1448","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/users\/503070"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1448\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/parts\/1445"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1448\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/media?parent=1448"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapter-type?post=1448"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/contributor?post=1448"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/license?post=1448"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}