For the following exercises (1-8), determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other.
1. [latex]f(x)=\dfrac{1}{\sqrt{x}}[/latex]
2. [latex]f(x)=\dfrac{2}{x^2+1}[/latex]
3. [latex]f(x)=\dfrac{x}{x^2-x}[/latex]
4. [latex]g(t)=t^{-1}+1[/latex]
5. [latex]f(x)=\dfrac{5}{e^x-2}[/latex]
6. [latex]f(x)=\dfrac{|x-2|}{x-2}[/latex]
7. [latex]H(x)= \tan 2x[/latex]
8. [latex]f(t)=\dfrac{t+3}{t^2+5t+6}[/latex]
For the following exercises (9-14), decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it?
9. [latex]f(x)=\dfrac{2x^2-5x+3}{x-1}[/latex] at [latex]x=1[/latex]
10. [latex]h(\theta)=\dfrac{\sin \theta - \cos \theta}{\tan \theta}[/latex] at [latex]\theta =\pi[/latex]
11. [latex]g(u)=\begin{cases} \dfrac{6u^2+u-2}{2u-1} & \text{ if } \, u \ne \frac{1}{2} \\ \dfrac{7}{2} & \text{ if } \, u = \frac{1}{2} \end{cases}[/latex] at [latex]u=\frac{1}{2}[/latex]
12. [latex]f(y)=\dfrac{\sin(\pi y)}{\tan(\pi y)}[/latex], at [latex]y=1[/latex]
13. [latex]f(x)=\begin{cases} x^2-e^x & \text{ if } \, x < 0 \\ x-1 & \text{ if } \, x \ge 0 \end{cases}[/latex] at [latex]x=0[/latex]
14. [latex]f(x)=\begin{cases} x \sin x & \text{ if } \, x \le \pi \\ x \tan x & \text{ if } \, x > \pi \end{cases}[/latex] at [latex]x=\pi[/latex]
In the following exercises (15-19), find the value(s) of [latex]k[/latex] that makes each function continuous over the given interval.
15. [latex]f(x)=\begin{cases} 3x+2 & \text{ if } \, x < k \\ 2x-3 & \text{ if } \, k \le x \le 8 \end{cases}[/latex]
16. [latex]f(\theta)=\begin{cases} \sin \theta & \text{ if } \, 0 \le \theta < \frac{\pi}{2} \\ \cos (\theta + k) & \text{ if } \, \frac{\pi}{2} \le \theta \le \pi \end{cases}[/latex]
17. [latex]f(x)=\begin{cases} \dfrac{x^2+3x+2}{x+2} & \text{ if } \, x \ne -2 \\ k & \text{ if } \, x = -2 \end{cases}[/latex]
18. [latex]f(x)=\begin{cases} e^{kx} & \text{ if } \, 0 \le x < 4 \\ x+3 & \text{ if } \, 4 \le x \le 8 \end{cases}[/latex]
19. [latex]f(x)=\begin{cases} \sqrt{kx} & \text{ if } \, 0 \le x \le 3 \\ x+1 & \text{ if } \, 3 < x \le 10 \end{cases}[/latex]
In the following exercises (20-21), use the Intermediate Value Theorem (IVT).
20. Let [latex]h(x)=\begin{cases} 3x^2-4 & \text{ if } \, x \le 2 \\ 5+4x & \text{ if } \, x > 2 \end{cases}[/latex] Over the interval [latex][0,4][/latex], there is no value of [latex]x[/latex] such that [latex]h(x)=10[/latex], although [latex]h(0)<10[/latex] and [latex]h(4)>10[/latex]. Explain why this does not contradict the IVT.
21. A particle moving along a line has at each time [latex]t[/latex] a position function [latex]s(t)[/latex], which is continuous. Assume [latex]s(2)=5[/latex] and [latex]s(5)=2[/latex]. Another particle moves such that its position is given by [latex]h(t)=s(t)-t[/latex]. Explain why there must be a value [latex]c[/latex] for [latex]2<c<5[/latex] such that [latex]h(c)=0[/latex].
22. [T] Use the statement “The cosine of [latex]t[/latex] is equal to [latex]t[/latex] cubed.”
- Write the statement as a mathematical equation.
- Prove that the equation in part (a) has at least one real solution.
- Use a calculator to find an interval of length 0.01 that contains a solution of the equation.
23. Apply the IVT to determine whether [latex]2^x=x^3[/latex] has a solution in one of the intervals [latex][1.25,1.375][/latex] or [latex][1.375,1.5][/latex]. Briefly explain your response for each interval.
24. Consider the graph of the function [latex]y=f(x)[/latex] shown in the following graph.
![A diagram illustrating the intermediate value theorem. There is a generic continuous curved function shown over the interval [a,b]. The points fa. and fb. are marked, and dotted lines are drawn from a, b, fa., and fb. to the points (a, fa.) and (b, fb.). A third point, c, is plotted between a and b. Since the function is continuous, there is a value for fc. along the curve, and a line is drawn from c to (c, fc.) and from (c, fc.) to fc., which is labeled as z on the y axis.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/2332/2018/01/11203520/CNX_Calc_Figure_02_04_201.jpg)
- Find all values for which the function is discontinuous.
- For each value in part (a), use the formal definition of continuity to explain why the function is discontinuous at that value.
- Classify each discontinuity as either jump, removable, or infinite.
25. Let [latex]f(x)=\begin{cases} 3x & \text{ if } \, x > 1 \\ x^3 & \text{ if } \, x < 1 \end{cases}[/latex]
- Sketch the graph of [latex]f[/latex].
- Is it possible to find a value [latex]k[/latex] such that [latex]f(1)=k[/latex], which makes [latex]f(x)[/latex] continuous for all real numbers? Briefly explain.
26. Let [latex]f(x)=\dfrac{x^4-1}{x^2-1}[/latex] for [latex]x\ne -1,1[/latex].
- Sketch the graph of [latex]f[/latex].
- Is it possible to find values [latex]k_1[/latex] and [latex]k_2[/latex] such that [latex]f(-1)=k_1[/latex] and [latex]f(1)=k_2[/latex], and that makes [latex]f(x)[/latex] continuous for all real numbers? Briefly explain.
27. Sketch the graph of the function [latex]y=f(x)[/latex] with properties 1 through 7.
- The domain of [latex]f[/latex] is [latex](−\infty,+\infty)[/latex].
- [latex]f[/latex] has an infinite discontinuity at [latex]x=-6[/latex].
- [latex]f(-6)=3[/latex]
- [latex]\underset{x\to -3^-}{\lim}f(x)=\underset{x\to -3^+}{\lim}f(x)=2[/latex]
- [latex]f(-3)=3[/latex]
- [latex]f[/latex] is left continuous but not right continuous at [latex]x=3[/latex].
- [latex]\underset{x\to -\infty}{\lim}f(x)=−\infty[/latex] and [latex]\underset{x\to +\infty}{\lim}f(x)=+\infty[/latex]
28. Sketch the graph of the function [latex]y=f(x)[/latex] with properties 1 through 4.
- The domain of [latex]f[/latex] is [latex][0,5][/latex].
- [latex]\underset{x\to 1^+}{\lim}f(x)[/latex] and [latex]\underset{x\to 1^-}{\lim}f(x)[/latex] exist and are equal.
- [latex]f(x)[/latex] is left continuous but not continuous at [latex]x=2[/latex], and right continuous but not continuous at [latex]x=3[/latex].
- [latex]f(x)[/latex] has a removable discontinuity at [latex]x=1[/latex], a jump discontinuity at [latex]x=2[/latex], and the following limits hold: [latex]\underset{x\to 3^-}{\lim}f(x)=−\infty[/latex] and [latex]\underset{x\to 3^+}{\lim}f(x)=2[/latex].
In the following exercises (29-30), suppose [latex]y=f(x)[/latex] is defined for all [latex]x[/latex]. For each description, sketch a graph with the indicated property.
29. Discontinuous at [latex]x=1[/latex] with [latex]\underset{x\to -1}{\lim}f(x)=-1[/latex] and [latex]\underset{x\to 2}{\lim}f(x)=4[/latex]
30. Discontinuous at [latex]x=2[/latex] but continuous elsewhere with [latex]\underset{x\to 0}{\lim}f(x)=\frac{1}{2}[/latex]
Determine whether each of the given statements is true (31-37). Justify your responses with an explanation or counterexample.
31. [latex]f(t)=\dfrac{2}{e^t-e^{-t}}[/latex] is continuous everywhere.
32. If the left- and right-hand limits of [latex]f(x)[/latex] as [latex]x\to a[/latex] exist and are equal, then [latex]f[/latex] cannot be discontinuous at [latex]x=a[/latex].
33. If a function is not continuous at a point, then it is not defined at that point.
34. According to the IVT, [latex]\cos x - \sin x - x = 2[/latex] has a solution over the interval [latex][-1,1][/latex].
35. If [latex]f(x)[/latex] is continuous such that [latex]f(a)[/latex] and [latex]f(b)[/latex] have opposite signs, then [latex]f(x)=0[/latex] has exactly one solution in [latex][a,b][/latex].
36. The function [latex]f(x)=\dfrac{x^2-4x+3}{x^2-1}[/latex] is continuous over the interval [latex][0,3][/latex].
37. If [latex]f(x)[/latex] is continuous everywhere and [latex]f(a), f(b)>0[/latex], then there is no root of [latex]f(x)[/latex] in the interval [latex][a,b][/latex].
The following problems (38-39) consider the scalar form of Coulomb’s law, which describes the electrostatic force between two point charges, such as electrons. It is given by the equation [latex]F(r)=k_e\dfrac{|q_1q_2|}{r^2}[/latex], where [latex]k_e[/latex] is Coulomb’s constant, [latex]q_i[/latex] are the magnitudes of the charges of the two particles, and [latex]r[/latex] is the distance between the two particles.
38. [T] To simplify the calculation of a model with many interacting particles, after some threshold value [latex]r=R[/latex], we approximate [latex]F[/latex] as zero.
- Explain the physical reasoning behind this assumption.
- What is the force equation?
- Evaluate the force [latex]F[/latex] using both Coulomb’s law and our approximation, assuming two protons with a charge magnitude of [latex]1.6022 \times 10^{-19} \, \text{coulombs (C)}[/latex], and the Coulomb constant [latex]k_e = 8.988 \times 10^9 \, \text{Nm}^2/\text{C}^2[/latex] are 1 m apart. Also, assume [latex]R<1\text{m}[/latex]. How much inaccuracy does our approximation generate? Is our approximation reasonable?
- Is there any finite value of [latex]R[/latex] for which this system remains continuous at [latex]R[/latex]?
39. [T] Instead of making the force 0 at [latex]R[/latex], instead we let the force be [latex]10^{-20}[/latex] for [latex]r\ge R[/latex]. Assume two protons, which have a magnitude of charge [latex]1.6022 \times 10^{-19} \, \text{C}[/latex], and the Coulomb constant [latex]k_e=8.988 \times 10^9 \, \text{Nm}^2/\text{C}^2[/latex]. Is there a value [latex]R[/latex] that can make this system continuous? If so, find it.
Recall the discussion on spacecraft from the Why It Matters. The following problems (40-42) consider a rocket launch from Earth’s surface. The force of gravity on the rocket is given by [latex]F(d)=\frac{-mk}{d^2}[/latex], where [latex]m[/latex] is the mass of the rocket, [latex]d[/latex] is the distance of the rocket from the center of Earth, and [latex]k[/latex] is a constant.
40. [T] Determine the value and units of [latex]k[/latex] given that the mass of the rocket on Earth is 3 million kg. (Hint: The distance from the center of Earth to its surface is 6378 km.)
41. [T] After a certain distance [latex]D[/latex] has passed, the gravitational effect of Earth becomes quite negligible, so we can approximate the force function by [latex]F(d)=\begin{cases} -\dfrac{mk}{d^2} & \text{ if } \, d < D \\ 10,000 & \text{ if } \, d \ge D \end{cases}[/latex] Find the necessary condition [latex]D[/latex] such that the force function remains continuous.
42. As the rocket travels away from Earth’s surface, there is a distance [latex]D[/latex] where the rocket sheds some of its mass, since it no longer needs the excess fuel storage. We can write this function as [latex]F(d)=\begin{cases} -\dfrac{m_1 k}{d^2} & \text{ if } \, d < D \\ -\dfrac{m_2 k}{d^2} & \text{ if } \, d \ge D \end{cases}[/latex] Is there a [latex]D[/latex] value such that this function is continuous, assuming [latex]m_1 \ne m_2[/latex]?
Prove the following functions are continuous everywhere (43-45).
43. [latex]f(\theta) = \sin \theta[/latex]
44. [latex]g(x)=|x|[/latex]
45. Where is [latex]f(x)=\begin{cases} 0 & \text{ if } \, x \, \text{is irrational} \\ 1 & \text{ if } \, x \, \text{is rational} \end{cases}[/latex] continuous?
