## Problem Set 72: Continuity

1. State in your own words what it means for a function $f$ to be continuous at $x=c$.

2. State in your own words what it means for a function to be continuous on the interval $\left(a,b\right)$.

For the following exercises, determine why the function $f$ is discontinuous at a given point $a$ on the graph. State which condition fails.

3. $f\left(x\right)=\mathrm{ln}\text{ }|\text{ }x+3\text{ }|,a=-3$

4. $f\left(x\right)=\mathrm{ln}\text{ }|\text{ }5x - 2\text{ }|,a=\frac{2}{5}$

5. $f\left(x\right)=\frac{{x}^{2}-16}{x+4},a=-4$

6. $f\left(x\right)=\frac{{x}^{2}-16x}{x},a=0$

7. $f\left(x\right)=\begin{cases}x,\hfill& x\neq 3 \\ 2x, \hfill& x=3\end{cases}a=3$

8. $f\left(x\right)=\begin{cases}5, \hfill& x\neq 0 \\ 3, \hfill& x=0\end{cases} a=0$

9. $f\left(x\right)=\begin{cases}\frac{1}{2-x}, \hfill& x\neq 2 \\ 3, \hfill& x=2\end{cases}a=2$

10. $f\left(x\right)=\begin{cases}\frac{1}{x+6}, \hfill& x=-6 \\ x^{2}, \hfill& x\neq -6\end{cases}a=-6$

11. $f\left(x\right)=\begin{cases}3+x, \hfill& x<1 \\ x, \hfill& x=1 \\ x^{2}, \hfill& x>1\end{cases}a=1$

12. $f\left(x\right)=\begin{cases}3-x, \hfill& x<1 \\ x, \hfill& x=1 \\ 2x^{2}, \hfill& x>1\end{cases} a=1$

13. $f\left(x\right)=\begin{cases}3+2x, \hfill& x<1 \\ x, \hfill& x=1 \\ -x^{2}, \hfill& x>1\end{cases}a=1$

14. $f\left(x\right)=\begin{cases}x^{2}, \hfill& x<-2 \\ 2x+1, \hfill& x=-2 \\ x^{3}, \hfill& x>-2\end{cases}a=-2$

15. $f\left(x\right)=\begin{cases}\frac{x^{2}-9}{x+3}, \hfill& x<-3 \\ x-9, \hfill& x=-3 \\ \frac{1}{x}, \hfill& x>-3\end{cases}a=-3$

16. $f\left(x\right)=\begin{cases}\frac{x^{2}-9}{x+3}, \hfill& x<-3 \\ x-9, \hfill& x=-3 \\ -6, \hfill& x>-3\end{cases}a=3$

17. $f\left(x\right)=\frac{{x}^{2}-4}{x - 2},\text{ }a=2$

18. $f\left(x\right)=\frac{25-{x}^{2}}{{x}^{2}-10x+25},\text{ }a=5$

19. $f\left(x\right)=\frac{{x}^{3}-9x}{{x}^{2}+11x+24},\text{ }a=-3$

20. $f\left(x\right)=\frac{{x}^{3}-27}{{x}^{2}-3x},\text{ }a=3$

21. $f\left(x\right)=\frac{x}{|x|},\text{ }a=0$

22. $f\left(x\right)=\frac{2|x+2|}{x+2},\text{ }a=-2$

For the following exercises, determine whether or not the given function $f$ is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.

23. $f\left(x\right)={x}^{3}-2x - 15$

24. $f\left(x\right)=\frac{{x}^{2}-2x - 15}{x - 5}$

25. $f\left(x\right)=2\cdot {3}^{x+4}$

26. $f\left(x\right)=\mathrm{-sin}\left(3x\right)$

27. $f\left(x\right)=\frac{|x - 2|}{{x}^{2}-2x}$

28. $f\left(x\right)=\tan \left(x\right)+2$

29. $f\left(x\right)=2x+\frac{5}{x}$

30. $f\left(x\right)={\mathrm{log}}_{2}\left(x\right)$

31. $f\left(x\right)=\mathrm{ln}\text{ }{x}^{2}$

32. $f\left(x\right)={e}^{2x}$

33. $f\left(x\right)=\sqrt{x - 4}$

34. $f\left(x\right)=\sec \left(x\right)-3$ .

35. $f\left(x\right)={x}^{2}+\sin \left(x\right)$

36. Determine the values of $b$ and $c$ such that the following function is continuous on the entire real number line.

$f\left(x\right)=\begin{cases}x+1, \hfill& {1 }<{x }<{3 }\\ x^{2}+bx+c, \hfill& |x-2|\geq 1\end{cases}$

For the following exercises, refer to Figure 15. Each square represents one square unit. For each value of $a$, determine which of the three conditions of continuity are satisfied at $x=a$ and which are not.

Figure 15

37. $x=-3$

38. $x=2$

39. $x=4$

For the following exercises, use a graphing utility to graph the function $f\left(x\right)=\sin \left(\frac{12\pi }{x}\right)$ as in Figure 16. Set the x-axis a short distance before and after 0 to illustrate the point of discontinuity.

Figure 16

40. Which conditions for continuity fail at the point of discontinuity?

41. Evaluate $f\left(0\right)$.

42. Solve for $x$ if $f\left(x\right)=0$.

43. What is the domain of $f\left(x\right)?$

For the following exercises, consider the function shown in Figure 17.

Figure 17

44. At what x-coordinates is the function discontinuous?

45. What condition of continuity is violated at these points?

46. Consider the function shown in Figure 18. At what x-coordinates is the function discontinuous? What condition(s) of continuity were violated?

Figure 18

47. Construct a function that passes through the origin with a constant slope of 1, with removable discontinuities at $x=-7$ and $x=1$.

48. The function $f\left(x\right)=\frac{{x}^{3}-1}{x - 1}$ is graphed in Figure 19. It appears to be continuous on the interval $\left[-3,3\right]$, but there is an x-value on that interval at which the function is discontinuous. Determine the value of $x$ at which the function is discontinuous, and explain the pitfall of utilizing technology when considering continuity of a function by examining its graph.

Figure 19

49. Find the limit $\underset{x\to 1}{\mathrm{lim}}f\left(x\right)$ and determine if the following function is continuous at $x=1:$

50. The function is discontinuous at $x=1$ because the limit as $x$ approaches 1 is 5 and $f\left(1\right)=2$.

51. The graph of $f\left(x\right)=\frac{\sin \left(2x\right)}{x}$ is shown in Figure 20. Is the function $f\left(x\right)$ continuous at $x=0?$ Why or why not?

Figure 20