Q: Which of the following is the correct definition of continuity for a function $$f(x) at a $$point $$x = c$$?

$$f(x) is $$continuous at $$x = c if$$ $$\lim\limits_{x \to c} f(x) = f(c)$$

Q: Which of the following statements about removable discontinuities is TRUE?

A removable discontinuity occurs when $$\lim\limits_{x \to c} f(x)$$ exists but $$f(c) is $$either not defined or not equal to the limit

Q: If $$f(x) is $$continuous from the right at $$x = c$$, which of the following must be true?

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

Q: Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} \frac{1}{x+1}, & x \neq -1 \\ 2, & x = -1 \end{cases}. $$What value should $$f(-1) be $$changed to in order to remove the discontinuity at $$x = -1$$?

$$f(-1)$$ should be changed to $$\lim\limits_{x \to -1} \frac{1}{x+1}$$, which does not exist, so the discontinuity is non-removable

Q: A function $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ + 1, & x \neq 0 \\ 2, & x = 0 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 0$$?

No, because $$\lim\limits_{x \to 0} f(x) = 1$$ but $$f(0) = 2$$

Question 1

Which of the following is the correct definition of continuity for a function $$f(x) at a $$point $$x = c$$?

Accepted Answer

$$f(x) is $$continuous at $$x = c if$$ $$\lim\limits_{x 	o c} f(x) = f(c)$$

Question 2

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ - 1, & x \leq -1 \ 2x + 1, & -1 \u003c x \u003c 2 \ x^2$ - 2x, & x \geq 2 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = -1$$?

Accepted Answer

No, because the left and right limits at $$x = -1$$ are not equal

Question 3

Given $$f(x) = \frac{\sin x}{x}$$, at which point is $$f(x)$$ not continuous?

Accepted Answer

$$x = 0$$

Question 4

Which of the following statements about removable discontinuities is TRUE?

Accepted Answer

A removable discontinuity occurs when $$\lim\limits_{x 	o c} f(x)$$ exists but $$f(c) is $$either not defined or not equal to the limit

Question 5

Let $$f(x) = \sqrt{x + 3}. $$For which values of $$x is$$ $$f(x)$$ continuous?

Accepted Answer

$$x \geq -3$$

Question 6

Suppose $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 2x}{x^2$ - 1}. At $$which $$x-$$values does $$f(x)$$ have a discontinuity?

Accepted Answer

$$x = 1$$ and $$x = -1$$

Question 7

Given $$f(x) = \frac{x^2$ - 2x}{x^2$ - 1}$$, which discontinuity is removable?

Accepted Answer

$$x = 1 is $$removable, $$x = -1 is $$non-removable

Question 8

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

Accepted Answer

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

Question 9

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ - 1, & x \leq -1 \ 2x + 1, & -1 \u003c x \u003c 2 \ x^2$ - 2x, & x \geq 2 \end{cases}. $$What value should be assigned to $$f(2) to $$make $$f(x)$$ continuous at $$x = 2$$?

Accepted Answer

$$f(2) = 0$$

Question 10

If $$f(x) is $$continuous from the right at $$x = c$$, which of the following must be true?

Accepted Answer

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

Question 11

Given $$f(x) = \frac{2x^2$ - 2x}{x^2$ - 1}$$, for which value of $$x is $$the discontinuity removable?

Accepted Answer

$$x = 1$$

Question 12

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} \frac{1}{x+1}, & x 
eq -1 \ 2, & x = -1 \end{cases}. $$What value should $$f(-1) be $$changed to in order to remove the discontinuity at $$x = -1$$?

Accepted Answer

$$f(-1)$$ should be changed to $$\lim\limits_{x 	o -1} \frac{1}{x+1}$$, which does not exist, so the discontinuity is non-removable

Question 13

A function $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 4}{x - 2}. $$What type of discontinuity does $$f(x)$$ have at $$x = 2$$?

Accepted Answer

Removable

Question 14

Which of the following combinations of continuous functions is NOT necessarily continuous?

Accepted Answer

$$\sqrt{f(x)}$$ where $$f(x) is $$continuous and $$f(x) \u003c 0$$ for some $$x$$

Question 15

Suppose $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 2x}{x^2$ - 1}. $$For which value of $$x$$ does $$f(x)$$ have a non-removable discontinuity?

Accepted Answer

$$x = -1$$

Question 16

A function $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ + 1, & x 
eq 0 \ 2, & x = 0 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 0$$?

Accepted Answer

No, because $$\lim\limits_{x 	o 0} f(x) = 1$$ but $$f(0) = 2$$

Question 17

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

Accepted Answer

There exists $$c in$$ $$(a, b)$$ such that $$f(c) = 0$$

Question 18

Suppose $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 4}{x - 2}. $$What value should be assigned to $$f(2) to $$make $$f(x)$$ continuous at $$x = 2$$?

Accepted Answer

$$f(2) = 4$$

Continuity of Functions and Types of Discontinuities

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