Here are our typical home and exam practice problems about Continuity and the Intermediate Value Theorem in Calculus, each of course with a complete solution a click away. It’s all free, with the goal of supporting your learning and helping you do well in your Calculus course.

**Update:** We now have *much* more interactive ways for you to learn about the important concept of “continuity,” making heavy use of Desmos graphing calculators so you can work with these ideas for yourself, and develop your problem solving skills step-by-step. Please visit our Continuity Section to *really* get this material down for yourself.

It’s all free, and waiting for you! (Why? Just because we’re educators who believe you deserve the chance to develop a better understanding of Calculus for yourself, and so we’re aiming to provide that. We hope you’ll take advantage!)

And if you’re just looking for typical problems and solutions to these types of problems, previous students have found what’s below super-helpful.

Question 1: Is *f* continuous? (Based on university exam problems)

Determine whether each of the following functions is continuous everywhere. If not, state the discontinuity or discontinuities.**(a)** $y = |x|$**(b)** $y = \dfrac{x}{x^2 + 1}$**(c)** $y = \dfrac{1 + x^2}{1 - x^2}$**(d)** $y = \sqrt{x^2 + 4}$**(e)** $y = x^{\frac{2}{3}}$**(f)** $y = \csc x$

Question 2: Based on an actual exam problem

Consider the function $f(x) = \sqrt{|x-1|}$.**(a)** What is the domain of $f(x)$? The range?**(b)** Graph the function. Make sure to label the axes.**(c)** Is $f(x)$ an even function, an odd function, or neither? Explain.**(d)** Find $\displaystyle{\lim_{x\to 1}f(x)}$ if it exists. If not, explain why not.**(e)** Is $f(x)$ continuous at $x=1$? Explain.

Question 3: True or False

Answer the following **True** or **False**. Explain your reasoning.**(a)** **True** or **False**? You were once exactly 2 feet tall.**(b)** At the end of the first half of a basketball game, a team has scored 24 points.

**True** or **False**? There must have been a moment during the first half when the team had exactly 15 points.

Question 4: Choose C to make *f* continuous (a common exam question)

[*Note:* This type of problem *frequently* appears on exams.]

Consider the function $$ f(x) = \begin{cases} \displaystyle{\frac{x^2 - 9}{x-3}} & \text{if } x \ne 3 \\ C & \text{if } x = 3 \end{cases} $$ For what value of $C$ is $f(x)$ continuous?

Consider the function $$ f(x) = \begin{cases} \displaystyle{\frac{x^2 - 9}{x-3}} & \text{if } x \ne 3 \\ C & \text{if } x = 3 \end{cases} $$ For what value of $C$ is $f(x)$ continuous?

Question 5: Discontinuity and asymptotes (based on an actual exam problem)

Consider $\displaystyle{f(x) = \frac{2x-2}{x^2 +2x - 3}}$.**(a)** (i) Find the values of *x*, if any, where *f(x)* is discontinuous. (ii) If any of these discontinuities are removable, state the value the function would need to have at that point to be continuous.**(b)** Write an equation for each vertical and horizontal asymptote of the graph of $f$.

Question 6: Show that *f* has a zero

Answer the following separate questions.**(a)** Show that the function $f(x) = -3x^3 - 2x^2 +3x +1$ has a zero between $x = -1$ and $x =0$.

*Note:* You may not use a calculator to answer this question.**(b)** Prove that $x = \cos{(x)}$ has at least one solution.

*Note:* You may not use a calculator to answer this question.

Question 7: Is there a number 1 more than its cube? (Actual university exam question)

Is there a number that is exactly 1 more than its cube? *Note:* You may not use a calculator to answer this question.

Question 8: 0 if *x* rational; 1 if *x* irrational

Consider the function
$$
f(x) =
\begin{cases}
0 & \text{if } x \text{ is rational} \\
1 & \text{if } x \text{ is irrational}
\end{cases}
$$
For what values of $x$ is $f$ continuous?

Question 9: Hiker on path

A hiker starts walking from the bottom of a mountain at 6:00 a.m., following a path, and arrives at the top of the mountain at 6:00 p.m. The next day she starts from the top at 6:00 a.m. and takes the same path to the bottom of the mountain, arriving at 6:00 p.m.

Prove using the intermediate value theorem that there is a point on the path that the hiker will cross at exactly the same time of the day on both days.

Prove using the intermediate value theorem that there is a point on the path that the hiker will cross at exactly the same time of the day on both days.

Question 10: Antipodal points and temperature

Two points on the surface of the Earth are called *antipodal* if they are at exactly opposite points. (For example, the North Pole and South Pole are antipodal points). Prove that, at any given moment, there are two antipodal points on the equator with exactly the same temperature. *Hint: *Let $T(\theta)$ be the temperature, at any given moment, at the point on the equator with longitudinal angle $\theta$ measured in radians, $0 \le \theta \le 2\pi$. (That is, in one complete trip around the equator, $\theta$ goes from 0 to $2\pi$.) Consider the function $f(\theta) = T(\theta + \pi) - T(\theta)$.

☕ *Buy us a coffee* If we've helped, please consider

giving a little something back.

Thank you! 😊

We'd love to hear:

- What questions do you have about the solutions above?
- Which ones are giving you the most trouble?
- What other continuity or IVT problems are you trying to work through for your class?

As of September 2022, we’re using our Forum for comments and discussion of this topic, and for any math questions. We’d love to see you there! Please tap to visit our Forum: community.matheno.com.

☕

Buy us a coffeeWe're working to add more,and would appreciate your help

to keep going! 😊