Q: For the piecewise function $$f(x) = \begin{cases} 1-x & x 0 \end{cases}$$, 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: Where is the function $$f(x) = \frac{\ln x + \tan$$^{-1}$$ x}{x^2$ - 1}$$ continuous?

$$x \u003e 0$$ and $$x \neq 1$$

Q: For the function $$f(x) = \frac{x^2$ - x - 6}{x - 3}$$ for $$x \neq 3$$ and $$f(x) = 2x - 5$$ for $$x = 3$$, is $$f(x)$$ continuous at $$x = 3$$?

No, because $$\lim\limits_{x \to 3} f(x) \neq f(3).$$

Q: A function $$f(x) is $$said to be right-continuous at $$x = c if$$:

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

Q: Which of the following is an example of a function with a jump discontinuity?

$$f(x) = \begin{cases} 1 & x \u003c 0 \\ 2 & x \geq 0 \end{cases}$$

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

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

Question 1

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

Accepted Answer

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

Question 2

Consider the function $$f(x) = \frac{1}{x^2$}. At $$which value of $$x is$$ $$f(x)$$ NOT continuous?

Accepted Answer

$$x = 0$$

Question 3

Which of the following types of functions is always continuous on its domain?

Accepted Answer

Polynomials

Question 4

Given $$f(x) = \frac{2x+3}{x-4}$$, at $$c = 2$$, is $$f(x)$$ continuous? Why or why not?

Accepted Answer

Yes, because $$f(2)$$ exists and the limit equals the function value.

Question 5

For the piecewise function $$f(x) = \begin{cases} 1-x & x  0 \end{cases}$$, 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 6

Where is the function $$f(x) = \frac{\ln x + 	an$$^{-1}$$ x}{x^2$ - 1}$$ continuous?

Accepted Answer

$$x \u003e 0$$ and $$x 
eq 1$$

Question 7

For the function $$f(x) = \frac{x^2$ - x - 6}{x - 3}$$ for $$x 
eq 3$$ and $$f(x) = 2x - 5$$ for $$x = 3$$, is $$f(x)$$ continuous at $$x = 3$$?

Accepted Answer

No, because $$\lim\limits_{x 	o 3} f(x) 
eq f(3).$$

Question 8

Which of the following best describes a removable discontinuity?

Accepted Answer

A point where the function is not defined but the limit exists.

Question 9

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} bx^2$ + 2x & x \u003c 2 \ x^3$ - bx & x \geq 2 \end{cases}. $$For what value of $$b is$$ $$f(x)$$ continuous everywhere?

Accepted Answer

$$b = 1$$

Question 10

A function $$f(x) is $$said to be right-continuous at $$x = c if$$:

Accepted Answer

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

Question 11

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

Accepted Answer

$$f(x) = \begin{cases} 1 & x \u003c 0 \ 2 & x \geq 0 \end{cases}$$

Question 12

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

Accepted Answer

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

Question 13

A real-world example: The temperature $$T(t) in a $$chemical reaction is measured over time. If $$T(t) is $$not continuous at $$t = 5$$ minutes, what could this indicate?

Accepted Answer

There was a sudden change in temperature at $$t = 5$$ minutes.

Question 14

Which of the following statements is TRUE about the function $$f(x) = \ln(x)$$?

Accepted Answer

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

Question 15

If $$f(x) is $$not continuous at $$x = c$$ because $$\lim\limits_{x 	o c} f(x)$$ does not exist, what type of discontinuity could this be?

Accepted Answer

Jump or infinite discontinuity

2.5

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