Q: Which of the following best describes a removable discontinuity at $x=a$ for a function $f(x)$?

The limit $\lim_{x \to a} f(x)$ exists, but $f(a) \neq \lim_{x \to a} f(x)$.

Q: Suppose $f(x)$ is continuous on $[a, b]$ and $f(a) < N < f(b)$. What does the Intermediate Value Theorem guarantee?

There exists $c$ in $(a, b)$ such that $f(c) = N$

Q: Which of the following is NOT a requirement for a function to be continuous at $x=a$?

$f(x)$ is differentiable at $x=a$

Q: Let $f(x) = \frac{x^2 - 4}{x - 2}$ for $x \neq 2$ and $f(2) = 4$. Is $f(x)$ continuous at $x=2$?

Yes, because $f(2) = \lim_{x \to 2} f(x)$

Q: A function $f(x)$ has an infinite discontinuity at $x=a$ if:

$\lim_{x \to a} f(x) = \pm \infty$

Q: If $f(x)$ is only continuous from the right at $a$ and only from the left at $b$, which statement is true?

$f(x)$ is not continuous at $a$ and $b$

Q: If $f(x)$ has a hole in its graph at $x=a$, which of the following is true?

$f(a)$ is not defined, but $\lim_{x \to a} f(x)$ exists$

Q: A function $f(x)$ is continuous on an interval $[a, b]$ if:

$f(x)$ is continuous at every point in $(a, b)$ and from the right at $a$ and from the left at $b$

Q: Which of the following is a real-world application of the Intermediate Value Theorem?

Guaranteeing that a chemical reaction reaches a certain temperature between two measured values

Question 1

Which of the following best describes a removable discontinuity at $x=a$ for a function $f(x)$?

Accepted Answer

The limit $\lim_{x 	o a} f(x)$ exists, but $f(a) 
eq \lim_{x 	o a} f(x)$.

Question 2

Given $f(x) = \frac{x^2 - x - 2}{x - 2}$ for $x 
eq 2$ and $f(2) = 1$, what type of discontinuity does $f(x)$ have at $x=2$?

Accepted Answer

Removable discontinuity

Question 3

For the function $f(x) = [x]$ (the greatest integer function), what type of discontinuity occurs at $x=1$?

Accepted Answer

Jump discontinuity

Question 4

Suppose $f(x)$ is defined as $f(x) = \frac{1}{x-4}$. What happens at $x=4$?

Accepted Answer

Infinite discontinuity

Question 5

Which of the following is TRUE about the Intermediate Value Theorem?

Accepted Answer

It guarantees a root for $f(x)$ in $[a, b]$ if $f(a)$ and $f(b)$ have opposite signs.

Question 6

Let $f(x) = \begin{cases} x^2 - 4 & x < 2 \ ax^2 - bx + 3 & x \geq 2 \end{cases}$. What condition must $a$ and $b$ satisfy for $f(x)$ to be continuous at $x=2$?

Accepted Answer

$4a - 2b + 3 = 4$

Question 7

A function $f$ is continuous from the right at $a$ if:

Accepted Answer

$\lim_{x 	o a^+} f(x) = f(a)$

Question 8

Consider $f(x) = \begin{cases} e^x & x \leq 0 \ 2 - x & x > 0 \end{cases}$. At $x=0$, is $f$ continuous, and if not, what type of discontinuity is present?

Accepted Answer

Jump discontinuity

Question 9

If $\lim_{x 	o a^-} f(x) = L$ and $\lim_{x 	o a^+} f(x) = M$ with $L 
eq M$, what type of discontinuity does $f(x)$ have at $x=a$?

Accepted Answer

Jump discontinuity

Question 10

Suppose $f(x)$ is continuous on $[a, b]$ and $f(a) < N < f(b)$. What does the Intermediate Value Theorem guarantee?

Accepted Answer

There exists $c$ in $(a, b)$ such that $f(c) = N$

Question 11

Which of the following is NOT a requirement for a function to be continuous at $x=a$?

Accepted Answer

$f(x)$ is differentiable at $x=a$

Question 12

Let $f(x) = \frac{x^2 - 4}{x - 2}$ for $x 
eq 2$ and $f(2) = 4$. Is $f(x)$ continuous at $x=2$?

Accepted Answer

Yes, because $f(2) = \lim_{x 	o 2} f(x)$

Question 13

A function $f(x)$ has an infinite discontinuity at $x=a$ if:

Accepted Answer

$\lim_{x 	o a} f(x) = \pm \infty$

Question 14

If $f(x)$ is only continuous from the right at $a$ and only from the left at $b$, which statement is true?

Accepted Answer

$f(x)$ is not continuous at $a$ and $b$

Question 15

Which of the following functions is continuous everywhere?

Accepted Answer

$f(x) = \sin x$

Question 16

Suppose $f(x)$ is defined as $f(x) = \begin{cases} x^2 & x < 1 \ 2x - 1 & x \geq 1 \end{cases}$. What must be true for $f(x)$ to be continuous at $x=1$?

Accepted Answer

$1^2 = 2 \cdot 1 - 1$

Question 17

If $f(x)$ has a hole in its graph at $x=a$, which of the following is true?

Accepted Answer

$f(a)$ is not defined, but $\lim_{x 	o a} f(x)$ exists$

Question 18

A function $f(x)$ is continuous on an interval $[a, b]$ if:

Accepted Answer

$f(x)$ is continuous at every point in $(a, b)$ and from the right at $a$ and from the left at $b$

Question 19

Which of the following is a real-world application of the Intermediate Value Theorem?

Accepted Answer

Guaranteeing that a chemical reaction reaches a certain temperature between two measured values

Continuity and Types of Discontinuity in Functions

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