Q: A function $$f(x) is $$defined as $$f(x) = \dfrac{x^2$ - x - 6}{x - 3}$$ for $$x \neq 3$$ and $$f(3) = 5. Is$$ $$f(x)$$ continuous at $$x = 3$$?

No, because $$\displaystyle \lim_{x \to 3} f(x) \neq f(3)$$

Q: Suppose $$f(x) is $$continuous from the right at $$x = a. $$Which of the following must be true?

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

Q: A function $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ & x \leq 1 \\ 2x + 1 & x \u003e 1 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 1$$?

Yes, because $$\displaystyle \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1)$$

Question 1

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

Accepted Answer

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

Question 2

A function $$f(x)$$ has a jump discontinuity at $$x = 2. $$Which of the following best describes this type of discontinuity?

Accepted Answer

The left and right limits at $$x = 2$$ exist but are not equal

Question 3

Given the function $$f(x) = \dfrac{\ln x + 	an$$^{-1}$$ x}{x^2$ - 1}$$, for which values of $$x is$$ $$f(x)$$ continuous?

Accepted Answer

$$(-\infty, -1) \cup (-1, 1) \cup (1, \infty)$$

Question 4

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} 1-x & x  0 \end{cases}$$ and $$f(0) = c. $$For which value of $$c is$$ $$f(x)$$ continuous at $$x = 0$$?

Accepted Answer

$$f(x)$$ cannot be made continuous at $$x = 0$$ for any $$c$$

Question 5

A function $$f(x) is $$defined as $$f(x) = \dfrac{x^2$ - x - 6}{x - 3}$$ for $$x 
eq 3$$ and $$f(3) = 5. Is$$ $$f(x)$$ continuous at $$x = 3$$?

Accepted Answer

No, because $$\displaystyle \lim_{x 	o 3} f(x) 
eq f(3)$$

Question 6

Which of the following types of functions is NOT guaranteed to be continuous everywhere in its domain?

Accepted Answer

Piecewise-defined functions

Question 7

A patient’s body temperature $$T(t) is $$modeled by a continuous function for $$t in $$hours. If $$T(t)$$ suddenly jumps from $$98.6^\circF to$$ $$102^\circF at$$ $$t = 5$$, what type of discontinuity does $$T(t)$$ have at $$t = 5$$?

Accepted Answer

Jump discontinuity

Question 8

Let $$f(x) = \begin{cases} x^3$ - bx & x \u003c 2 \ 8 - 2b & x \geq 2 \end{cases}. $$For which value of $$b is$$ $$f(x)$$ continuous at $$x = 2$$?

Accepted Answer

$$b = \dfrac{2}{3}$$

Question 9

A chemical reaction’s rate $$R(t) is $$modeled by $$R(t) = \dfrac{t^2$ - 4}{t - 2}$$ for $$t 
eq 2. $$What value should $$R(2) be $$assigned to make $$R(t)$$ continuous at $$t = 2$$?

Accepted Answer

$$2$$

Question 10

Which of the following statements about removable discontinuities is TRUE?

Accepted Answer

The limit exists at the point, but the function is either not defined or not equal to the limit

Question 11

Suppose $$f(x) is $$continuous from the right at $$x = a. $$Which of the following must be true?

Accepted Answer

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

Question 12

A function $$g(x) is $$defined as $$g(x) = \sqrt{x-1}$$ for $$x \geq 1. On $$which interval is $$g(x)$$ continuous?

Accepted Answer

$$[1, \infty)$$

Question 13

A function $$h(x) is $$continuous everywhere except at $$x = 1$$, where $$\displaystyle \lim_{x 	o 1^-} h(x) = 2$$ and $$\displaystyle \lim_{x 	o 1^+} h(x) = 5. $$What type of discontinuity does $$h(x)$$ have at $$x = 1$$?

Accepted Answer

Jump

Question 14

If $$f(x) is $$continuous on $$(-\infty, 3) \cup (3, \infty)$$ but not at $$x = 3$$, which of the following is the correct interval notation for the set of points where $$f(x) is $$continuous?

Accepted Answer

$$(-\infty, 3) \cup (3, \infty)$$

Question 15

A function $$f(x) is $$defined as $$f(x) = \dfrac{1}{x-4}. At $$which value of $$x$$ does $$f(x)$$ have an infinite discontinuity?

Accepted Answer

$$x = 4$$

Question 16

A function $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ & x \leq 1 \ 2x + 1 & x \u003e 1 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 1$$?

Accepted Answer

Yes, because $$\displaystyle \lim_{x 	o 1^-} f(x) = \lim_{x 	o 1^+} f(x) = f(1)$$

Question 17

A function $$f(x) is $$continuous everywhere except at $$x = 2$$, where $$f(2) is $$not defined. What type of discontinuity does $$f(x)$$ have at $$x = 2$$?

Accepted Answer

Removable

Question 18

Suppose $$f(x) is $$continuous for all $$x$$ except at $$x = 0$$, where $$\displaystyle \lim_{x 	o 0^-} f(x) = 1$$ and $$\displaystyle \lim_{x 	o 0^+} f(x) = 2. $$Which of the following is true?

Accepted Answer

$$f(x)$$ has a jump discontinuity at $$x = 0$$

Question 19

A function $$f(x) is $$defined as $$f(x) = \dfrac{x^2$ - 1}{x - 1}$$ for $$x 
eq 1. $$What value should $$f(1) be $$assigned to make $$f(x)$$ continuous at $$x = 1$$?

Accepted Answer

$$2$$

Continuity in Calculus: Definitions, Types, and Applications

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