Q: A function $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ - 36 \over x - 6 & x \neq 6 \\ 7 & x = 6 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 6$$?

No, because $$\lim\limits_{x \to 6} f(x) \neq f(6)$$

Q: For the function $$f(x) = \begin{cases} x^2$ + 1 & x \leq 0 \\ 3x + 5 & x \u003e 0 \end{cases}$$, is $$f(x)$$ continuous at $$x = 0$$?

No, because $$\lim\limits_{x \to 0^-} f(x) \neq \lim\limits_{x \to 0^+} f(x)$$

Q: Which of the following best describes right-continuity at an endpoint $$a$$?

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

Q: Which of the following is an example of a removable discontinuity?

A function where $$\lim\limits_{x \to a} f(x)$$ exists but $$f(a) is $$not defined

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

No, because $$\lim\limits_{x \to 0} f(x) \neq f(0)$$

Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

$$x = 1$$

Question 3

A function $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ - 36 \over x - 6 & x 
eq 6 \ 7 & x = 6 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 6$$?

Accepted Answer

No, because $$\lim\limits_{x 	o 6} f(x) 
eq f(6)$$

Question 4

For the function $$f(x) = \begin{cases} x^2$ + 1 & x \leq 0 \ 3x + 5 & x \u003e 0 \end{cases}$$, is $$f(x)$$ continuous at $$x = 0$$?

Accepted Answer

No, because $$\lim\limits_{x 	o 0^-} f(x) 
eq \lim\limits_{x 	o 0^+} f(x)$$

Question 5

Which of the following best describes right-continuity at an endpoint $$a$$?

Accepted Answer

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

Question 6

Determine the interval(s) of continuity for $$f(x) = \dfrac{x^2$ - 5x + 6}{x^2$ - 4}.$$

Accepted Answer

$$(-\infty, -2) \cup (-2, 2) \cup (2, \infty)$$

Question 7

For $$f(x) = \sqrt{x - 4}$$, what is the domain of continuity?

Accepted Answer

$$[4, \infty)$$

Question 8

A function $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ + 3x & x \u003e 2 \ 4x & x \leq 2 \end{cases}. At $$which value is $$f(x)$$ NOT continuous?

Accepted Answer

$$x = 2$$

Question 9

Suppose a function $$f(x) is $$continuous on $$[a, b). $$Which of the following statements is true?

Accepted Answer

$$f(x) is $$right-continuous at $$a$$

Question 10

A patient’s medication concentration in the bloodstream is modeled by $$C(t) = \dfrac{t^2$ - 4}{t - 2}$$ for $$t 
eq 2. $$What is the limit of $$C(t) as$$ $$t 	o 2$$?

Accepted Answer

$$2$$

Question 11

A chemical reaction rate is given by $$R(x) = \dfrac{x^2$ - 9}{x - 3}$$ for $$x 
eq 3. If$$ $$R(3) is $$defined as $$k$$, for which value of $$k is$$ $$R(x)$$ continuous at $$x = 3$$?

Accepted Answer

$$k = 6$$

Question 12

Which of the following is an example of a removable discontinuity?

Accepted Answer

A function where $$\lim\limits_{x 	o a} f(x)$$ exists but $$f(a) is $$not defined

Question 13

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

Accepted Answer

Yes, because both one-sided limits and $$f(1)$$ are equal

Question 14

A function $$f(x) is $$continuous everywhere except at $$x = 3$$, where $$\lim\limits_{x 	o 3^-} f(x) = 2$$ and $$\lim\limits_{x 	o 3^+} f(x) = 5. $$What type of discontinuity is this?

Accepted Answer

Jump

Question 15

Suppose $$f(x) is $$defined as $$f(x) = \dfrac{1}{x - 4}. On $$which interval is $$f(x)$$ continuous?

Accepted Answer

$$(-\infty, 4) \cup (4, \infty)$$

Question 16

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

Accepted Answer

No, because $$\lim\limits_{x 	o 0} f(x) 
eq f(0)$$

Question 17

A function $$f(x) is $$continuous on $$[1, 5). $$Which of the following must be true?

Accepted Answer

$$f(x) is $$right-continuous at $$x = 1$$

Question 18

A function $$f(x) is $$defined as $$f(x) = \dfrac{x^2$ - 4x + 3}{x - 3}. $$What is the limit as $$x 	o 3$$?

Accepted Answer

$$2$$

Question 19

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

Accepted Answer

No, because $$\lim\limits_{x 	o 2^-} f(x) 
eq \lim\limits_{x 	o 2^+} f(x)$$

Continuity: Definitions, Examples, and Interval Analysis 2.6

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