Q: Which of the following is true about removable discontinuities?

They occur when $$\lim_{x \to c} f(x)$$ exists but $$f(c) is $$not defined or not equal to the limit.

Question 1

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

Accepted Answer

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

Question 2

Given the function $$f(x) = \begin{cases} x^2$ + 3, & x \u003c 2 \ ax + 2, & x \geq 2 \end{cases}$$, what value of $$a$$ makes $$f(x)$$ continuous at $$x = 2$$?

Accepted Answer

$$a = 2$$

Question 3

Which type of discontinuity is present at $$x = 1 in $$the graph where $$\lim_{x 	o 1} f(x)$$ exists but $$f(1) is $$not defined?

Accepted Answer

Removable discontinuity

Question 4

Suppose $$f(x) is $$continuous on $$[0, 2]$$ and $$f(0) = -1$$, $$f(2) = 3. $$According to the Intermediate Value Theorem, which statement is true?

Accepted Answer

There exists $$c \in [0,2]$$ such that $$f(c) = 0$$

Question 5

Given $$f(x) = \frac{x^2$ - 4}{x - 2}$$, what type of discontinuity occurs at $$x = 2$$?

Accepted Answer

Removable discontinuity

Question 6

A function $$f(x) is $$defined as $$f(x) = \frac{1}{x-3}. $$What type of discontinuity does $$f(x)$$ have at $$x = 3$$?

Accepted Answer

Infinite discontinuity

Question 7

If $$\lim_{x 	o 5^-} f(x) = 2$$ and $$\lim_{x 	o 5^+} f(x) = 7$$, what type of discontinuity is present at $$x = 5$$?

Accepted Answer

Jump discontinuity

Question 8

Which property of continuous functions allows you to conclude that if $$f(x)$$ and $$g(x)$$ are continuous at $$x = c$$, then $$f(x) + g(x) is $$also continuous at $$x = c$$?

Accepted Answer

Sum property

Question 9

Let $$f(x) be $$continuous on $$[a, b]$$ and $$f(a) \u003c k \u003c f(b). $$What does the Intermediate Value Theorem guarantee?

Accepted Answer

There exists $$c \in [a, b]$$ such that $$f(c) = k$$

Question 10

Given $$f(x) = \begin{cases} x^2$, & x \u003c 3 \ 2x + c, & x \geq 3 \end{cases}$$, what value of $$c$$ makes $$f(x)$$ continuous at $$x = 3$$?

Accepted Answer

$$c = -3$$

Question 11

A patient’s blood concentration of a drug is modeled by a continuous function $$C(t)$$ over time $$t. If$$ $$C(2) = 1 mg/L $$and $$C(6) = 5 mg/L$$, what does the Intermediate Value Theorem guarantee?

Accepted Answer

There is a time $$t in$$ $$[2,6]$$ when $$C(t) = 3 mg/L$$

Question 12

Suppose $$f(x) is $$continuous everywhere except at $$x = 4$$, where $$\lim_{x 	o 4} f(x)$$ does not exist. What type of discontinuity is this?

Accepted Answer

Infinite discontinuity

Question 13

If $$f(x) is $$continuous at $$x = 0$$ and $$g(x) is $$continuous at $$x = 0$$, which of the following is NOT necessarily continuous at $$x = 0$$?

Accepted Answer

$$\frac{f(x)}{g(x)} if$$ $$g(0) = 0$$

Question 14

A chemical reaction rate $$R(t) is $$modeled by a continuous function. If $$R(0) = 2$$ and $$R(10) = 8$$, which value is guaranteed to occur at least once for some $$t in$$ $$[0,10]$$?

Accepted Answer

$$R(t) = 5$$

Question 15

Which of the following is true about removable discontinuities?

Accepted Answer

They occur when $$\lim_{x 	o c} f(x)$$ exists but $$f(c) is $$not defined or not equal to the limit.

Question 16

Given $$f(x) = \frac{4 \sin x}{x}$$, what is $$\lim_{x 	o 0} f(x)$$?

Accepted Answer

$$4$$

Question 17

If $$f(x) is $$continuous on $$[a, b]$$ and $$f(a)$$ and $$f(b)$$ have opposite signs, what does the Intermediate Value Theorem guarantee?

Accepted Answer

There is at least one $$c \in [a, b]$$ such that $$f(c) = 0$$

Question 18

A function $$f(x) is $$defined as $$f(x) = \begin{cases} x+2, & x \u003c 1 \ 3x-1, & x \geq 1 \end{cases}. $$What value of $$x is $$required for $$f(x) to be $$continuous at $$x = 1$$?

Accepted Answer

$$x = 1$$

Continuity and Discontinuity in Calculus: Definitions, Theorems, and Applications

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