{"id":863,"date":"2021-09-03T17:25:23","date_gmt":"2021-09-03T17:25:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=863"},"modified":"2022-10-29T01:17:39","modified_gmt":"2022-10-29T01:17:39","slug":"continuity","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/continuity\/","title":{"raw":"Continuity","rendered":"Continuity"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>State the conditions for continuity of a function of two variables.<\/li>\r\n \t<li>Verify the continuity of a function of two variables at a point.<\/li>\r\n \t<li>Calculate the limit of a function of three or more variables and verify the continuity of the function at a point.<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn <a href=\"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/introduction-to-continuity\/\" rel=\"noopener\" data-book-uuid=\"8b89d172-2927-466f-8661-01abc7ccdba4\" data-page-slug=\"2-4-continuity\">Continuity<\/a>, we defined the continuity of a function of one variable and saw how it relied on the limit of a function of one variable. In particular, three conditions are necessary for [latex]f\\,(x)[\/latex]\u00a0to be continuous at point [latex]x=a[\/latex]:\r\n<ol>\r\n \t<li>[latex]f\\,(a)[\/latex] exists.<\/li>\r\n \t<li>[latex]\\displaystyle{\\lim_{x\\to{a}}}f\\,(x)[\/latex] exists.<\/li>\r\n \t<li>[latex]\\displaystyle{\\lim_{x\\to{a}}}f\\,(x)=f\\,(a)[\/latex].<\/li>\r\n<\/ol>\r\nThese three conditions are necessary for continuity of a function of two variables as well.\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\r\n\r\n<hr \/>\r\n\r\nA function [latex]f\\,(x,\\ y)[\/latex] is continuous at a point [latex](a,\\ b)[\/latex]\u00a0in its domain if the following conditions are satisfied:\r\n<ol>\r\n \t<li>[latex]f\\,(a,\\ b)[\/latex] exists.<\/li>\r\n \t<li>[latex]\\displaystyle{\\lim_{(x,\\ y)\\to{(a,\\ b)}}}f\\,(x,\\ y)[\/latex] exists.<\/li>\r\n \t<li>[latex]\\displaystyle{\\lim_{(x,\\ y)\\to(a,\\ b)}}f\\,(x,\\ y)=f\\,(a,\\ b)[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Demonstrating Continuity for a Function of Two Variables<\/h3>\r\nShow that the function [latex]f\\,(x,\\ y)=\\frac{3x+2y}{x+y+1}[\/latex] is continuous at point [latex](5,-3)[\/latex].\r\n\r\n[reveal-answer q=\"fs-id1667793753114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1667793753114\"]\r\n<div style=\"text-align: left;\">There are three conditions to be satisfied, per the definition of continuity. In this example, [latex]a=5[\/latex] and [latex]b=-3[\/latex].<\/div>\r\n<ol>\r\n \t<li>[latex]f\\,(a,\\ b)[\/latex]\u00a0exists. This is true because the domain of the function [latex]f[\/latex]\u00a0consists of those ordered pairs for which the denominator is nonzero (i.e., [latex]x+y+1\\neq{0}[\/latex]). Point [latex](5,-3)[\/latex]\u00a0satisfies this condition. Furthermore,\r\n<div style=\"text-align: center;\">[latex]f\\,(a,\\ b)=f\\,(5,-3)=\\frac{3(5)+2(-3)}{5+(-3)+1}=\\frac{15-6}{2+1}=3.[\/latex]<\/div>\r\n&nbsp;<\/li>\r\n \t<li>[latex]\\displaystyle\\lim_{(x,\\ y)\\to(a,\\ b)}f\\,(x,\\ y)[\/latex] exists. This is also true:\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\displaystyle\\lim_{(x,\\ y)\\to(a,\\ b)}f\\,(x,\\ y)} &amp; =\\hfill &amp; {\\displaystyle\\lim_{(x,\\ y)\\to(5,\\ -3)}\\frac{3x+2y}{x+y+1}}\\hfill \\\\ \\hfill &amp; =\\hfill &amp; {\\frac{\\displaystyle\\lim_{(x,\\ y)\\to(5,\\ -3)}(3x+2y)}{\\displaystyle\\lim_{(x,\\ y)\\to(5,\\ -3)}(x+y+1)}}\\hfill \\\\ \\hfill &amp; =\\hfill &amp; {\\frac{15-6}{5-3+1}}\\hfill \\\\ \\hfill &amp; =\\hfill &amp; {3.}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div><\/li>\r\n \t<li>[latex]\\displaystyle\\lim_{(x,\\ y)\\to(a,\\ b)}f\\,(x,\\ y)=f\\,(a,\\ b)[\/latex].\u00a0This is true because we have just shown that both sides of this equation equal three.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\nShow that the function [latex]f\\,(x,y)=\\sqrt{26-2x^{2}-y^{2}}[\/latex]\u00a0is continuous at point [latex](2,-3)[\/latex].\r\n\r\n[reveal-answer q=\"fs-id1667993733114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1667993733114\"]\r\n<ol>\r\n \t<li style=\"text-align: left;\">The domain of [latex]f[\/latex] contains the ordered pair [latex](2,-3)[\/latex] because [latex]f\\,(a,b)=f\\,(2,-3)=\\sqrt{16-2(2)^{2}-(-3)^{2}}=3[\/latex]<\/li>\r\n \t<li>[latex]\\displaystyle\\lim_{(x,y)\\to(a,b)}f\\,(x,y)=3[\/latex]<\/li>\r\n \t<li>[latex]\\displaystyle\\lim_{(x,y)\\to(a,b)}f\\,(x,y)=f\\,(a,b)=3[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nContinuity of a function of any number of variables can also be defined in terms of delta and epsilon. A function of two variables is continuous at a point [latex](x_0,y_0)[\/latex] in its domain for every [latex]\\epsilon&gt;0[\/latex] there exists a [latex]\\delta&gt;0[\/latex] such that, whenever [latex]\\sqrt{(x-x_0)^2+(y-y_0)^2}&lt;\\delta[\/latex] it is true, [latex]|f(x,y)-f(a,b)|&lt;\\epsilon[\/latex]. This definition can be combined with the formal definition (that is, the\u00a0<em>epsilon-delta\u00a0<\/em><i>definition<\/i>) of continuity of a function of one variable to prove the following theorems:\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\" data-type=\"title\">the sum of continuous functions is continuous<\/h3>\r\n\r\n<hr \/>\r\n\r\nIf [latex]f(x,y)[\/latex] is continuous at\u00a0[latex](x_0,y_0)[\/latex], and\u00a0[latex]g(x,y)[\/latex] is continuous at\u00a0[latex](x_0,y_0)[\/latex], then\u00a0[latex]f(x,y)+g(x,y)[\/latex] is continuous at\u00a0[latex](x_0,y_0)[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\" data-type=\"title\">the product of continuous functions is continuous<\/h3>\r\n\r\n<hr \/>\r\n\r\nIf [latex]g(x)[\/latex] is continuous at\u00a0[latex]x_0[\/latex], and\u00a0[latex]h(y)[\/latex] is continuous at\u00a0[latex]y_0[\/latex], then\u00a0[latex]f(x,y)=g(x)h(y)[\/latex] is continuous at\u00a0[latex](x_0,y_0)[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\" data-type=\"title\">the composition of continuous functions is continuous<\/h3>\r\n\r\n<hr \/>\r\n\r\nLet [latex]g[\/latex] be a function of two variables from a domain\u00a0[latex]D\\subseteq\\mathbb{R}^{2}[\/latex] to a range [latex]R\\subseteq\\mathbb{R}[\/latex]. Suppose\u00a0[latex]g[\/latex] is continuous at some point\u00a0[latex](x_0,y_0)\\in{D}[\/latex] and define [latex]z_0=g(x_0,y_0)[\/latex]. Let [latex]f[\/latex] be a function that maps\u00a0[latex]\\mathbb{R}[\/latex] to\u00a0[latex]\\mathbb{R}[\/latex] such that [latex]z_0[\/latex] is in the domain of [latex]f[\/latex]. Last, assume [latex]f[\/latex] is continuous at [latex]z_0[\/latex]. Then [latex]f\\circ{g}[\/latex] is continuous at [latex](x_0,y_0)[\/latex] as shown in the following figure.\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_979\" align=\"aligncenter\" width=\"856\"]<img class=\"size-full wp-image-979\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/28225710\/4-2-7.jpeg\" alt=\"A shape is shown labeled the domain of g with point (x, y) inside of it. From the domain of g there is an arrow marked g pointing to the range of g, which is a straight line with point z on it. The range of g is also marked the domain of f. Then there is another arrow marked f from this shape to a line marked range of f.\" width=\"856\" height=\"228\" \/> Figure 1.\u00a0The composition of two continuous functions is continuous.[\/caption]\r\n\r\nLet\u2019s now use the previous theorems to show continuity of functions in the following examples.\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0More Examples of Continuity of a Function of Two Variables<\/h3>\r\nShow that the functions [latex]f\\,(x,\\ y)=4x^{3}y^{2}[\/latex] and [latex]g\\,(x,\\ y)=\\cos{(4x^{3}y^{2})}[\/latex] are continuous everywhere.\r\n\r\n[reveal-answer q=\"fs-id2667993733114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id2667993733114\"]\r\n\r\nThe polynomials [latex]g\\,(x)=4x^{3}[\/latex] and [latex]h\\,(y)=y^{2}[\/latex]\u00a0are continuous at every real number, and therefore by the product of continuous functions theorem, [latex]f\\,(x,\\ y)=4x^{3}y^{2}[\/latex] is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane. Since [latex]f\\,(x,\\ y)=4x^{3}y^{2}[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane and [latex]g\\,(x)=\\cos{x}[\/latex]\u00a0is continuous at every real number [latex]x[\/latex],\u00a0the continuity of the composition of functions tells us that [latex]g\\,(x,\\ y)=\\cos{(4x^{3}y^{2})}[\/latex] is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\nShow that the functions [latex]f\\,(x,\\ y)=2x^{2}y^{3}+3[\/latex]\u00a0and [latex]g\\,(x,\\ y)=(2x^{2}y^{3}+3)^{4}[\/latex] are continuous everywhere.\r\n\r\n[reveal-answer q=\"fs-id1687993733114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1687993733114\"]\r\n\r\nThe polynomials [latex]g\\,(x)=2x^{2}[\/latex] and [latex]h\\,(y)=y^{3}[\/latex]\u00a0are continuous at every real number; therefore, by the product of continuous functions theorem, [latex]f\\,(x,\\ y)=2x^{2}y^{3}[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane.\u00a0Furthermore, any constant function is continuous everywhere, so [latex]g\\,(x,\\ y)=3[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane. Therefore, [latex]f\\,(x,\\ y)=2x^{2}y^{3}[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane. Last, [latex]h\\,(x)=x^{4}[\/latex]\u00a0is continuous at every real number [latex]x[\/latex],\u00a0so by the continuity of composite functions theorem [latex]g\\,(x,\\ y)=(2x^{2}y^{3}+3)^{4}[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to the above Try It[\/caption]\r\n\r\n<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8186152&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=k4SalFB3XEw&amp;video_target=tpm-plugin-310u6ibm-k4SalFB3XEw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.10_transcript.html\">transcript for \u201cCP 4.10\u201d here (opens in new window).<\/a><\/center>\r\n<h2>Functions of Three or More Variables<\/h2>\r\nThe limit of a function of three or more variables occurs readily in applications. For example, suppose we have a function [latex]f\\,(x,\\ y,\\ z)[\/latex]\u00a0that gives the temperature at a physical location [latex](x,\\ y,\\ z)[\/latex]\u00a0in three dimensions. Or perhaps a function [latex]g\\,(x,\\ y,\\ z,\\ t)[\/latex]\u00a0can indicate air pressure at a location\u00a0[latex](x,\\ y,\\ z)[\/latex] at time [latex]t[\/latex]. How can we take a limit at a point in [latex]\\mathbb{R}^{3}[\/latex]?\u00a0What does it mean to be continuous at a point in four dimensions?\r\n<p id=\"fs-id1167793488748\" class=\" \">The answers to these questions rely on extending the concept of a [latex]\\delta[\/latex]\u00a0disk into more than two dimensions. Then, the ideas of the limit of a function of three or more variables and the continuity of a function of three or more variables are very similar to the definitions given earlier for a function of two variables.<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\r\n\r\n<hr \/>\r\n\r\nLet [latex](x_0,\\ y_0,\\ z_0)[\/latex] be a point in [latex]\\mathbb{R}^{3}[\/latex]. Then, a [latex]\\delta[\/latex]\u00a0<strong>ball<\/strong>\u00a0in three dimensions consists of all points in [latex]\\mathbb{R}^{3}[\/latex]\u00a0lying at a distance of less than [latex]\\delta[\/latex] from\u00a0[latex](x_0,\\ y_0,\\ z_0)[\/latex] - that is,\r\n<div style=\"text-align: center;\">[latex]\\large{\\big\\{(x,\\ y,\\ z)\\in\\mathbb{R}^{3}\\big|\\sqrt{(x-x_0)^{2}+(y-y_0)^{2}+(z-z_0)^{2}}\\,&lt;\\delta\\big\\}}[\/latex].<\/div>\r\n&nbsp;\r\n\r\nTo define a [latex]\\delta[\/latex]\u00a0ball in higher dimensions, add additional terms under the radical to correspond to each additional dimension. For example, given a point [latex]P=(w_0,\\ x_0,\\ y_0,\\ z_0)[\/latex] in [latex]\\mathbb{R}^{4}[\/latex], a [latex]\\delta[\/latex] ball around [latex]P[\/latex] can be described by\r\n<div style=\"text-align: center;\">[latex]\\large{\\big\\{(w,\\ x,\\ y,\\ z)\\in\\mathbb{R}^{4}\\big|\\sqrt{(w-w_0)^{2}+(x-x_0)^{2}+(y-y_0)^{2}+(z-z_0)^{2}}\\,&lt;\\delta\\big\\}}[\/latex].<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\nTo show that a limit of a function of three variables exists at a point [latex](x_0,\\ y_0,\\ z_0)[\/latex],\u00a0it suffices to show that for any point in a [latex]\\delta[\/latex] ball centered at [latex](x_0,\\ y_0,\\ z_0)[\/latex],\u00a0the value of the function at that point is arbitrarily close to a fixed value (the limit value). All the limit laws for functions of two variables hold for functions of more than two variables as well.\r\n<div class=\"textbox exercises\">\r\n<h3>Example:\u00a0Finding the Limit of a Function of Three Variables<\/h3>\r\nFind [latex]\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}\\frac{x^{2}y-3z}{2x+5y-z}.[\/latex]\r\n\r\n[reveal-answer q=\"fs-id1647993733114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1647993733114\"]\r\n\r\nBefore we can apply the quotient law, we need to verify that the limit of the denominator is nonzero. Using the difference law, the identity law, and the constant law,\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}(2x+5y-z)} &amp; =\\hfill &amp; {2\\left(\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}x\\right)+5\\left(\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}y\\right)-\\left(\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}z\\right)} \\hfill \\\\ \\hfill &amp;=\\hfill &amp; {2(4)+5(1)-(-3)} \\hfill \\\\ \\hfill &amp; =\\hfill &amp; {16.}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n\r\nSince this\u00a0is nonzero, we next find the limit of the numerator. Using the product law, difference law, constant multiple law, and identity law,\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}(x^{2}y-3z)} &amp; =\\hfill &amp; {\\left(\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}x\\right)^{2}\\,\\left(\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}y\\right)-3\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}z}\\hfill \\\\ \\hfill &amp; =\\hfill &amp; {(4^{2})(1)-3(-3)}\\hfill \\\\ \\hfill &amp; =\\hfill &amp; {16+9}\\hfill \\\\ \\hfill &amp; =\\hfill &amp; {25.}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n\r\nLast, applying the quotient law:\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}\\frac{x^{2}y-3z}{2x+5y-z}} &amp;=\\hfill &amp; {\\frac{\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}(x^{2}y-3z)}{\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}(2x+5y-z)}}\\hfill \\\\ \\hfill &amp; =\\hfill &amp; {\\frac{25}{16}.} \\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\nFind [latex]\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ -1,\\ 3)}\\sqrt{13-x^{2}-2y^{2}+z^{2}}[\/latex].\r\n\r\n[reveal-answer q=\"fs-id1687993733119\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1687993733119\"]\r\n<div style=\"text-align: center;\">[latex]\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ -1,\\ 3)}\\sqrt{13-x^{2}-2y^{2}+z^{2}}=2[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>State the conditions for continuity of a function of two variables.<\/li>\n<li>Verify the continuity of a function of two variables at a point.<\/li>\n<li>Calculate the limit of a function of three or more variables and verify the continuity of the function at a point.<\/li>\n<\/ul>\n<\/div>\n<p>In <a href=\"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/introduction-to-continuity\/\" rel=\"noopener\" data-book-uuid=\"8b89d172-2927-466f-8661-01abc7ccdba4\" data-page-slug=\"2-4-continuity\">Continuity<\/a>, we defined the continuity of a function of one variable and saw how it relied on the limit of a function of one variable. In particular, three conditions are necessary for [latex]f\\,(x)[\/latex]\u00a0to be continuous at point [latex]x=a[\/latex]:<\/p>\n<ol>\n<li>[latex]f\\,(a)[\/latex] exists.<\/li>\n<li>[latex]\\displaystyle{\\lim_{x\\to{a}}}f\\,(x)[\/latex] exists.<\/li>\n<li>[latex]\\displaystyle{\\lim_{x\\to{a}}}f\\,(x)=f\\,(a)[\/latex].<\/li>\n<\/ol>\n<p>These three conditions are necessary for continuity of a function of two variables as well.<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\n<hr \/>\n<p>A function [latex]f\\,(x,\\ y)[\/latex] is continuous at a point [latex](a,\\ b)[\/latex]\u00a0in its domain if the following conditions are satisfied:<\/p>\n<ol>\n<li>[latex]f\\,(a,\\ b)[\/latex] exists.<\/li>\n<li>[latex]\\displaystyle{\\lim_{(x,\\ y)\\to{(a,\\ b)}}}f\\,(x,\\ y)[\/latex] exists.<\/li>\n<li>[latex]\\displaystyle{\\lim_{(x,\\ y)\\to(a,\\ b)}}f\\,(x,\\ y)=f\\,(a,\\ b)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Demonstrating Continuity for a Function of Two Variables<\/h3>\n<p>Show that the function [latex]f\\,(x,\\ y)=\\frac{3x+2y}{x+y+1}[\/latex] is continuous at point [latex](5,-3)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1667793753114\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1667793753114\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: left;\">There are three conditions to be satisfied, per the definition of continuity. In this example, [latex]a=5[\/latex] and [latex]b=-3[\/latex].<\/div>\n<ol>\n<li>[latex]f\\,(a,\\ b)[\/latex]\u00a0exists. This is true because the domain of the function [latex]f[\/latex]\u00a0consists of those ordered pairs for which the denominator is nonzero (i.e., [latex]x+y+1\\neq{0}[\/latex]). Point [latex](5,-3)[\/latex]\u00a0satisfies this condition. Furthermore,\n<div style=\"text-align: center;\">[latex]f\\,(a,\\ b)=f\\,(5,-3)=\\frac{3(5)+2(-3)}{5+(-3)+1}=\\frac{15-6}{2+1}=3.[\/latex]<\/div>\n<p>&nbsp;<\/li>\n<li>[latex]\\displaystyle\\lim_{(x,\\ y)\\to(a,\\ b)}f\\,(x,\\ y)[\/latex] exists. This is also true:\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\displaystyle\\lim_{(x,\\ y)\\to(a,\\ b)}f\\,(x,\\ y)} & =\\hfill & {\\displaystyle\\lim_{(x,\\ y)\\to(5,\\ -3)}\\frac{3x+2y}{x+y+1}}\\hfill \\\\ \\hfill & =\\hfill & {\\frac{\\displaystyle\\lim_{(x,\\ y)\\to(5,\\ -3)}(3x+2y)}{\\displaystyle\\lim_{(x,\\ y)\\to(5,\\ -3)}(x+y+1)}}\\hfill \\\\ \\hfill & =\\hfill & {\\frac{15-6}{5-3+1}}\\hfill \\\\ \\hfill & =\\hfill & {3.}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li>[latex]\\displaystyle\\lim_{(x,\\ y)\\to(a,\\ b)}f\\,(x,\\ y)=f\\,(a,\\ b)[\/latex].\u00a0This is true because we have just shown that both sides of this equation equal three.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p>Show that the function [latex]f\\,(x,y)=\\sqrt{26-2x^{2}-y^{2}}[\/latex]\u00a0is continuous at point [latex](2,-3)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1667993733114\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1667993733114\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li style=\"text-align: left;\">The domain of [latex]f[\/latex] contains the ordered pair [latex](2,-3)[\/latex] because [latex]f\\,(a,b)=f\\,(2,-3)=\\sqrt{16-2(2)^{2}-(-3)^{2}}=3[\/latex]<\/li>\n<li>[latex]\\displaystyle\\lim_{(x,y)\\to(a,b)}f\\,(x,y)=3[\/latex]<\/li>\n<li>[latex]\\displaystyle\\lim_{(x,y)\\to(a,b)}f\\,(x,y)=f\\,(a,b)=3[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>Continuity of a function of any number of variables can also be defined in terms of delta and epsilon. A function of two variables is continuous at a point [latex](x_0,y_0)[\/latex] in its domain for every [latex]\\epsilon>0[\/latex] there exists a [latex]\\delta>0[\/latex] such that, whenever [latex]\\sqrt{(x-x_0)^2+(y-y_0)^2}<\\delta[\/latex] it is true, [latex]|f(x,y)-f(a,b)|<\\epsilon[\/latex]. This definition can be combined with the formal definition (that is, the\u00a0<em>epsilon-delta\u00a0<\/em><i>definition<\/i>) of continuity of a function of one variable to prove the following theorems:<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\" data-type=\"title\">the sum of continuous functions is continuous<\/h3>\n<hr \/>\n<p>If [latex]f(x,y)[\/latex] is continuous at\u00a0[latex](x_0,y_0)[\/latex], and\u00a0[latex]g(x,y)[\/latex] is continuous at\u00a0[latex](x_0,y_0)[\/latex], then\u00a0[latex]f(x,y)+g(x,y)[\/latex] is continuous at\u00a0[latex](x_0,y_0)[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\" data-type=\"title\">the product of continuous functions is continuous<\/h3>\n<hr \/>\n<p>If [latex]g(x)[\/latex] is continuous at\u00a0[latex]x_0[\/latex], and\u00a0[latex]h(y)[\/latex] is continuous at\u00a0[latex]y_0[\/latex], then\u00a0[latex]f(x,y)=g(x)h(y)[\/latex] is continuous at\u00a0[latex](x_0,y_0)[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\" data-type=\"title\">the composition of continuous functions is continuous<\/h3>\n<hr \/>\n<p>Let [latex]g[\/latex] be a function of two variables from a domain\u00a0[latex]D\\subseteq\\mathbb{R}^{2}[\/latex] to a range [latex]R\\subseteq\\mathbb{R}[\/latex]. Suppose\u00a0[latex]g[\/latex] is continuous at some point\u00a0[latex](x_0,y_0)\\in{D}[\/latex] and define [latex]z_0=g(x_0,y_0)[\/latex]. Let [latex]f[\/latex] be a function that maps\u00a0[latex]\\mathbb{R}[\/latex] to\u00a0[latex]\\mathbb{R}[\/latex] such that [latex]z_0[\/latex] is in the domain of [latex]f[\/latex]. Last, assume [latex]f[\/latex] is continuous at [latex]z_0[\/latex]. Then [latex]f\\circ{g}[\/latex] is continuous at [latex](x_0,y_0)[\/latex] as shown in the following figure.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div id=\"attachment_979\" style=\"width: 866px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-979\" class=\"size-full wp-image-979\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/09\/28225710\/4-2-7.jpeg\" alt=\"A shape is shown labeled the domain of g with point (x, y) inside of it. From the domain of g there is an arrow marked g pointing to the range of g, which is a straight line with point z on it. The range of g is also marked the domain of f. Then there is another arrow marked f from this shape to a line marked range of f.\" width=\"856\" height=\"228\" \/><\/p>\n<p id=\"caption-attachment-979\" class=\"wp-caption-text\">Figure 1.\u00a0The composition of two continuous functions is continuous.<\/p>\n<\/div>\n<p>Let\u2019s now use the previous theorems to show continuity of functions in the following examples.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0More Examples of Continuity of a Function of Two Variables<\/h3>\n<p>Show that the functions [latex]f\\,(x,\\ y)=4x^{3}y^{2}[\/latex] and [latex]g\\,(x,\\ y)=\\cos{(4x^{3}y^{2})}[\/latex] are continuous everywhere.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id2667993733114\">Show Solution<\/span><\/p>\n<div id=\"qfs-id2667993733114\" class=\"hidden-answer\" style=\"display: none\">\n<p>The polynomials [latex]g\\,(x)=4x^{3}[\/latex] and [latex]h\\,(y)=y^{2}[\/latex]\u00a0are continuous at every real number, and therefore by the product of continuous functions theorem, [latex]f\\,(x,\\ y)=4x^{3}y^{2}[\/latex] is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane. Since [latex]f\\,(x,\\ y)=4x^{3}y^{2}[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane and [latex]g\\,(x)=\\cos{x}[\/latex]\u00a0is continuous at every real number [latex]x[\/latex],\u00a0the continuity of the composition of functions tells us that [latex]g\\,(x,\\ y)=\\cos{(4x^{3}y^{2})}[\/latex] is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p>Show that the functions [latex]f\\,(x,\\ y)=2x^{2}y^{3}+3[\/latex]\u00a0and [latex]g\\,(x,\\ y)=(2x^{2}y^{3}+3)^{4}[\/latex] are continuous everywhere.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1687993733114\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1687993733114\" class=\"hidden-answer\" style=\"display: none\">\n<p>The polynomials [latex]g\\,(x)=2x^{2}[\/latex] and [latex]h\\,(y)=y^{3}[\/latex]\u00a0are continuous at every real number; therefore, by the product of continuous functions theorem, [latex]f\\,(x,\\ y)=2x^{2}y^{3}[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane.\u00a0Furthermore, any constant function is continuous everywhere, so [latex]g\\,(x,\\ y)=3[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane. Therefore, [latex]f\\,(x,\\ y)=2x^{2}y^{3}[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane. Last, [latex]h\\,(x)=x^{4}[\/latex]\u00a0is continuous at every real number [latex]x[\/latex],\u00a0so by the continuity of composite functions theorem [latex]g\\,(x,\\ y)=(2x^{2}y^{3}+3)^{4}[\/latex]\u00a0is continuous at every point [latex](x,\\ y)[\/latex] in the [latex]xy[\/latex]-plane.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try It<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8186152&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=k4SalFB3XEw&amp;video_target=tpm-plugin-310u6ibm-k4SalFB3XEw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP4.10_transcript.html\">transcript for \u201cCP 4.10\u201d here (opens in new window).<\/a><\/div>\n<h2>Functions of Three or More Variables<\/h2>\n<p>The limit of a function of three or more variables occurs readily in applications. For example, suppose we have a function [latex]f\\,(x,\\ y,\\ z)[\/latex]\u00a0that gives the temperature at a physical location [latex](x,\\ y,\\ z)[\/latex]\u00a0in three dimensions. Or perhaps a function [latex]g\\,(x,\\ y,\\ z,\\ t)[\/latex]\u00a0can indicate air pressure at a location\u00a0[latex](x,\\ y,\\ z)[\/latex] at time [latex]t[\/latex]. How can we take a limit at a point in [latex]\\mathbb{R}^{3}[\/latex]?\u00a0What does it mean to be continuous at a point in four dimensions?<\/p>\n<p id=\"fs-id1167793488748\" class=\"\">The answers to these questions rely on extending the concept of a [latex]\\delta[\/latex]\u00a0disk into more than two dimensions. Then, the ideas of the limit of a function of three or more variables and the continuity of a function of three or more variables are very similar to the definitions given earlier for a function of two variables.<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\n<hr \/>\n<p>Let [latex](x_0,\\ y_0,\\ z_0)[\/latex] be a point in [latex]\\mathbb{R}^{3}[\/latex]. Then, a [latex]\\delta[\/latex]\u00a0<strong>ball<\/strong>\u00a0in three dimensions consists of all points in [latex]\\mathbb{R}^{3}[\/latex]\u00a0lying at a distance of less than [latex]\\delta[\/latex] from\u00a0[latex](x_0,\\ y_0,\\ z_0)[\/latex] &#8211; that is,<\/p>\n<div style=\"text-align: center;\">[latex]\\large{\\big\\{(x,\\ y,\\ z)\\in\\mathbb{R}^{3}\\big|\\sqrt{(x-x_0)^{2}+(y-y_0)^{2}+(z-z_0)^{2}}\\,<\\delta\\big\\}}[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p>To define a [latex]\\delta[\/latex]\u00a0ball in higher dimensions, add additional terms under the radical to correspond to each additional dimension. For example, given a point [latex]P=(w_0,\\ x_0,\\ y_0,\\ z_0)[\/latex] in [latex]\\mathbb{R}^{4}[\/latex], a [latex]\\delta[\/latex] ball around [latex]P[\/latex] can be described by<\/p>\n<div style=\"text-align: center;\">[latex]\\large{\\big\\{(w,\\ x,\\ y,\\ z)\\in\\mathbb{R}^{4}\\big|\\sqrt{(w-w_0)^{2}+(x-x_0)^{2}+(y-y_0)^{2}+(z-z_0)^{2}}\\,<\\delta\\big\\}}[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<p>To show that a limit of a function of three variables exists at a point [latex](x_0,\\ y_0,\\ z_0)[\/latex],\u00a0it suffices to show that for any point in a [latex]\\delta[\/latex] ball centered at [latex](x_0,\\ y_0,\\ z_0)[\/latex],\u00a0the value of the function at that point is arbitrarily close to a fixed value (the limit value). All the limit laws for functions of two variables hold for functions of more than two variables as well.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example:\u00a0Finding the Limit of a Function of Three Variables<\/h3>\n<p>Find [latex]\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}\\frac{x^{2}y-3z}{2x+5y-z}.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1647993733114\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1647993733114\" class=\"hidden-answer\" style=\"display: none\">\n<p>Before we can apply the quotient law, we need to verify that the limit of the denominator is nonzero. Using the difference law, the identity law, and the constant law,<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}(2x+5y-z)} & =\\hfill & {2\\left(\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}x\\right)+5\\left(\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}y\\right)-\\left(\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}z\\right)} \\hfill \\\\ \\hfill &=\\hfill & {2(4)+5(1)-(-3)} \\hfill \\\\ \\hfill & =\\hfill & {16.}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>Since this\u00a0is nonzero, we next find the limit of the numerator. Using the product law, difference law, constant multiple law, and identity law,<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}(x^{2}y-3z)} & =\\hfill & {\\left(\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}x\\right)^{2}\\,\\left(\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}y\\right)-3\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}z}\\hfill \\\\ \\hfill & =\\hfill & {(4^{2})(1)-3(-3)}\\hfill \\\\ \\hfill & =\\hfill & {16+9}\\hfill \\\\ \\hfill & =\\hfill & {25.}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p>Last, applying the quotient law:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ccc}\\hfill {\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}\\frac{x^{2}y-3z}{2x+5y-z}} &=\\hfill & {\\frac{\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}(x^{2}y-3z)}{\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}(2x+5y-z)}}\\hfill \\\\ \\hfill & =\\hfill & {\\frac{25}{16}.} \\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p>Find [latex]\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ -1,\\ 3)}\\sqrt{13-x^{2}-2y^{2}+z^{2}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1687993733119\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1687993733119\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ -1,\\ 3)}\\sqrt{13-x^{2}-2y^{2}+z^{2}}=2[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\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-863\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>CP 4.10. <strong>Authored by<\/strong>: Ryan Melton. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/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":349141,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"CP 4.10\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"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-863","chapter","type-chapter","status-publish","hentry"],"part":22,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/863","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/349141"}],"version-history":[{"count":78,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/863\/revisions"}],"predecessor-version":[{"id":6446,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/863\/revisions\/6446"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/22"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/863\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=863"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=863"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=863"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=863"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}