📐 Continuity and the Intermediate Value Theorem

Questions in this quiz

Answers hidden — play to find out.

1. Question 1

   A function $f$ is continuous at $x = a$ if and only if which of the following is true?

2. Question 2

   The function $f(x) = \frac{x^2 - 4}{x - 2}$ is continuous at $x = 2$.

3. Question 3

   Which of the following functions has a jump discontinuity?

4. Question 4

   The Intermediate Value Theorem states that if $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then:

5. Question 5

   Consider $f(x) = x^3 - 3x + 1$ on the interval $[-2, 2]$. Which statement about the IVT is correct?

6. Question 6

   A removable discontinuity at $x = a$ occurs when $\lim_{x \to a} f(x)$ exists but ___ either does not exist or does not equal the limit.

7. Question 7

   Which of the following is a correct application of the Intermediate Value Theorem?

8. Question 8

   Let $f(x) = \begin{cases} x^2 & \text{if } x \leq 1 \\ 2x - 1 & \text{if } x > 1 \end{cases}$. At which point is $f$ discontinuous?

9. Question 9

   Arrange these steps in the correct order to apply the Intermediate Value Theorem to show that $f(x) = x^3 + x - 2$ has a solution between $x = 0$ and $x = 2$:

10. Question 10

    Suppose $f$ is continuous on $[a, b]$, $f(a) = 7$, and $f(b) = -3$. Which of the following statements is NOT necessarily true?