Q: Let $$f(x) = \begin{cases} x^2$, & x \u003c 1 \\ 2x-1, & x \geq 1 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 1$$?

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

Question 1

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

Accepted Answer

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

Question 2

A parking meter only accepts quarters, charging $$0.25$$ for every $$15$$ minutes of parking. Which type of discontinuity does the cost function for parking time exhibit?

Accepted Answer

Jump discontinuity

Question 3

Given $$f(x) = \frac{x^2$ - 4}{x - 2}$$, at $$x = 2$$ the function has:

Accepted Answer

A removable discontinuity

Question 4

Which of the following functions is continuous for all real numbers?

Accepted Answer

$$f(x) = \sin(x)$$

Question 5

Suppose $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) \u003c 0 \u003c f(b). $$What theorem guarantees that there is a $$c in$$ $$(a, b)$$ such that $$f(c) = 0$$?

Accepted Answer

Intermediate Value Theorem

Question 6

For which value(s) of $$x is $$the function $$f(x) = \frac{x^2$ - 9}{x - 3}$$ not continuous?

Accepted Answer

$$x = 3$$

Question 7

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

Accepted Answer

$$f(g(x)) is $$continuous at $$x = 2$$

Question 8

Which of the following best describes a removable discontinuity?

Accepted Answer

The limit exists but does not equal the function value

Question 9

Consider $$f(x) = \sqrt{x-1}. $$For which interval is $$f(x)$$ continuous?

Accepted Answer

$$[1, \infty)$$

Question 10

Which of the following statements about the function $$f(x) = 	an(x) is $$true?

Accepted Answer

$$f(x) is $$continuous except at $$x = \frac{\pi}{2} + n\pi, n \in \mathbb{Z}$$

Question 11

Let $$f(x) = \begin{cases} x^2$, & x \u003c 1 \ 2x-1, & x \geq 1 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 1$$?

Accepted Answer

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

Question 12

Which of the following is an example of a function with an infinite discontinuity?

Accepted Answer

$$f(x) = \frac{1}{x} at$$ $$x = 0$$

Question 13

If $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) = f(b)$$, which of the following must be true?

Accepted Answer

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

Question 14

Which of the following is true about the continuity of $$f(x) = \ln(x)$$?

Accepted Answer

$$f(x) is $$continuous for $$x \u003e 0$$

Question 15

Suppose $$f(x) is $$defined as $$f(x) = \frac{1}{x^2$ - 4}. At $$which points is $$f(x)$$ discontinuous?

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 16

If $$f(x) is $$continuous at $$x = 0$$ and $$f(0) = 5$$, what is $$\lim\limits_{x 	o 0} f(x)$$?

Accepted Answer

$$5$$

Question 17

Which of the following is a necessary condition for the continuity of $$f(x) = \sqrt{x} at$$ $$x = 0$$?

Accepted Answer

$$x \geq 0$$

Question 18

If $$f(x)$$ and $$g(x)$$ are both continuous at $$x = a$$, which of the following is always continuous at $$x = a$$?

Accepted Answer

All of the above

Question 19

Which of the following is true about the continuity of exponential functions $$f(x) = a^x$$?

Accepted Answer

$$f(x) is $$continuous for all real $$x$$

Continuity in Calculus: Definitions, Theorems, and Applications

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