$$h(x) = \begin{cases} x \sin\left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}$$

$$\lim\limits_{x \to a^+} f(x) = f(a)$$

Question 1

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

Accepted Answer

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

Question 2

Given $$f(x) = \dfrac{3x^2$ + 2x + 1}{x - 1}$$, at which value of $$x is$$ $$f(x)$$ NOT continuous?

Accepted Answer

$$x = 1$$

Question 3

Which of the following statements about polynomial functions is TRUE?

Accepted Answer

Polynomial functions are continuous at every real number.

Question 4

For the rational function $$f(x) = \dfrac{x}{x^2$ - 7x + 12}$$, at which values of $$x is$$ $$f(x)$$ discontinuous?

Accepted Answer

$$x = 3$$ and $$x = 4$$

Question 5

Which of the following functions is continuous at $$x = 0$$?

Accepted Answer

$$h(x) = \begin{cases} x \sin\left(\frac{1}{x}ight), & x 
eq 0 \ 0, & x = 0 \end{cases}$$

Question 6

If $$f$$ and $$g$$ are continuous at $$x = a$$, which of the following is NOT necessarily continuous at $$x = a$$?

Accepted Answer

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

Question 7

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

Accepted Answer

If $$f is $$continuous on $$[a, b]$$ and $$L is $$between $$f(a)$$ and $$f(b)$$, then there exists $$c in$$ $$(a, b)$$ such that $$f(c) = L.$$

Question 8

For which values of $$x is$$ $$g(x) = \sqrt{9 - x^2$}$$ continuous?

Accepted Answer

$$[-3, 3]$$

Question 9

Suppose $$f(x) is $$continuous on $$[1, 5]$$ and $$f(1) = -2$$, $$f(5) = 6. $$Which of the following must be true?

Accepted Answer

There exists $$c in$$ $$(1, 5)$$ such that $$f(c) = 0.$$

Question 10

Which of the following functions is continuous at all points of its domain?

Accepted Answer

$$\sin x$$

Question 11

Evaluate $$\lim\limits_{x 	o 1} \sqrt{2x^2$ - 1}.$$

Accepted Answer

$$\sqrt{1}$$

Question 12

Which of the following describes a function that is right-continuous at $$x = a$$?

Accepted Answer

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

Question 13

For which values of $$x is$$ $$f(x) = ((x^2$ - 2x + 4$$)^{1/3}$$)^2$ continuous?

Accepted Answer

For all real $$x$$

Question 14

If $$f is $$continuous at $$x = 2$$ and $$g is $$continuous at $$x = 2$$, which of the following is also continuous at $$x = 2$$?

Accepted Answer

$$f(x) + g(x)$$

Question 15

Which of the following is a correct statement about the continuity of composite functions?

Accepted Answer

If $$g is $$continuous at $$a$$ and $$f is $$continuous at $$g(a)$$, then $$f \circ g is $$continuous at $$a.$$

Question 16

Evaluate $$\lim\limits_{x 	o 2} \cos\left(\dfrac{x^2$ - 4}{x - 2}ight).$$

Accepted Answer

$$\cos(4)$$

Question 17

Which of the following is NOT a transcendental function that is continuous at all points of its domain?

Accepted Answer

$$	an x$$

Question 18

If $$f is $$continuous on $$[a, b]$$ and $$f(a) \u003c 0 \u003c f(b)$$, what does the Intermediate Value Theorem guarantee?

Accepted Answer

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

2.6 Continuity of Functions

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