2.1: Continuity

1. Sketch the graph of a continuous function that passes through the points $(-3, -1)$, $(-1, 2)$, $(0, -1)$, and $(2, 3)$.
    
2. Sketch the graph of a function that has a removable discontinuity at $x = 1$.
    
3. Sketch the graph of a function that has a jump discontinuity at $x = 2$.
    
4. Sketch the graph of a function that has an infinite discontinuity at $x = -1$.
    
5. Sketch the graph of the function $f(x) = x^2 - 1$. For what values of $x$ is the function continuous? For what values of $x$ is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
    
6. Sketch the graph of the function $f(x) = \dfrac{1}{x}$. For what values of $x$ is the function continuous? For what values of $x$ is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
    
7. Sketch the graph of the function $g(x) = \dfrac{1}{x^2 - 1}$. For what values of $x$ is the function continuous? For what values of $x$ is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
    
8. Sketch the graph of the function $h(x) = \dfrac{1}{x^2 + 1}$. For what values of $x$ is the function continuous? For what values of $x$ is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
    
9. Sketch the graph of the function $f(x) = \dfrac{x^2}{x}$. For what values of $x$ is the function continuous? For what values of $x$ is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
    
10. Sketch the graph of the function $g(x) = \dfrac{x+2}{x^2-x-6}$. For what values of $x$ is the function continuous? For what values of $x$ is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
     
11. Sketch the graph of the function $h(x) = \tan(x)$. For what values of $x$ is the function continuous? For what values of $x$ is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
     
12. Sketch the graph of the function $f(x) = \dfrac{|x|}{x}$. For what values of $x$ is the function continuous? For what values of $x$ is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
     
13. Sketch the graph of the function $g(x) = \dfrac{1}{\sqrt{x}}$. For what values of $x$ is the function continuous? For what values of $x$ is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
     
14. Sketch the graph of the function $h(x) = \sqrt{x^2-1}$. For what values of $x$ is the function continuous? For what values of $x$ is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
     
15. Sketch the graph of the function
    $$f(x) = \left\{\begin{align*} & x^2, & x \neq -1, \\ & 2, & x = -1. \end{align*}\right.$$
    For what values of $x$ is the function continuous? For what values of $x$ is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
     
16. Sketch the graph of the function
    $$g(x) = \left\{\begin{align*} & -x, & x < 0, \\ & \sqrt{x}, & x \ge 0. \end{align*}\right.$$
    For what values of $x$ is the function continuous? For what values of $x$ is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
     
17. Sketch the graph of the function
    $$h(x) = \left\{\begin{align*} & 2^x, & x \le 0, \\ & \frac{1}{x}, & x > 0. \end{align*}\right.$$
    For what values of $x$ is the function continuous? For what values of $x$ is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
     
18. Sketch the graph of the function
    $$h(x) = \left\{\begin{align*} & \frac{1}{x^2}, & x < 2, \\ & \pi, & x = 2, \\ & 2x-5, & x > 2. \end{align*}\right.$$
    For what values of $x$ is the function continuous? For what values of $x$ is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
     
19. Sketch the graph of the function $f(x) = \sin\left(\dfrac{1}{x}\right)$. For what values of $x$ is the function continuous? For what values of $x$ is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
     
20. Consider the function $g(x) = x^2$, and recall that $\sqrt{2} = 1.414213562\ldots$. The $x$-values in the table below approximate $x = \sqrt{2}$. The $x$-values on the left side of the table approach $\sqrt{2}$ from the left; the $x$-values on the right right side of the table approach $\sqrt{2}$ from the right. Use a calculator to complete the table by evaluating the function at these values. Do the function values give a good approximation of $g(\sqrt{2})$? Why or why not?
     

    $x$	$g(x)$		$x$	$g(x)$
    1.41			1.42
    1.414			1.415
    1.4142			1.4143

     

21. Consider the function $h(x) = \dfrac{x-1}{x^2-3x+2}$, and recall that $\sqrt{2} = 1.414213562\ldots$. The $x$-values in the table below approximate $x = \sqrt{2}$. The $x$-values on the left side of the table approach $\sqrt{2}$ from the left; the $x$-values on the right right side of the table approach $\sqrt{2}$ from the right. Use a calculator to complete the table by evaluating the function at these values. Do the function values give a good approximation of $h(\sqrt{2})$? Why or why not?
     

    $x$	$h(x)$		$x$	$h(x)$
    1.41			1.42
    1.414			1.415
    1.4142			1.4143

     

22. Consider the function $f(x) = \dfrac{|x^2 - 2|}{x^2 - 2}$, and recall that $\sqrt{2} = 1.414213562\ldots$. The $x$-values in the table below approximate $x = \sqrt{2}$. The $x$-values on the left side of the table approach $\sqrt{2}$ from the left; the $x$-values on the right right side of the table approach $\sqrt{2}$ from the right. Use a calculator to complete the table by evaluating the function at these values. Do the function values give a good approximation of $f(\sqrt{2})$? Why or why not?
     

    $x$	$f(x)$		$x$	$f(x)$
    1.41			1.42
    1.414			1.415
    1.4142			1.4143

     

23. Consider the function $g(x) = \dfrac{1}{x^2 - 2}$, and recall that $\sqrt{2} = 1.414213562\ldots$. The $x$-values in the table below approximate $x = \sqrt{2}$. The $x$-values on the left side of the table approach $\sqrt{2}$ from the left; the $x$-values on the right right side of the table approach $\sqrt{2}$ from the right. Use a calculator to complete the table by evaluating the function at these values. Do the function values give a good approximation of $g(\sqrt{2})$? Why or why not?
     

    $x$	$g(x)$		$x$	$g(x)$
    1.41			1.42
    1.414			1.415
    1.4142			1.4143

     

24. Consider the function $h(x) = \dfrac{x^2 - 2}{x^4 - 4}$, and recall that $\sqrt{2} = 1.414213562\ldots$. The $x$-values in the table below approximate $x = \sqrt{2}$. The $x$-values on the left side of the table approach $\sqrt{2}$ from the left; the $x$-values on the right right side of the table approach $\sqrt{2}$ from the right. Use a calculator to complete the table by evaluating the function at these values. Do the function values give a good approximation of $h(\sqrt{2})$? Why or why not?
     

    $x$	$h(x)$		$x$	$h(x)$
    1.41			1.42
    1.414			1.415
    1.4142			1.4143

     

25. Consider the function $f(x) = \dfrac{1}{x^2} - 2$. Evaluate $f(1)$ and $f(2)$. Does the Intermediate Value Theorem guarantee that the function has a zero on the interval $(1, 2)$? Why or why not? Confirm your answer using a graphing calculator.
     

26. Consider the function $g(x) = \dfrac{1}{x^2 - 2}$. Evaluate $g(1)$ and $g(2)$. Does the Intermediate Value Theorem guarantee that the function has a zero on the interval $(1, 2)$? Why or why not? Confirm your answer using a graphing calculator.