Q: A function $$f(x)$$ has a jump discontinuity at $$x = 2. $$Which of the following best describes this situation?

The left and right limits as $$x \to 2$$ are not equal

Q: Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$, & x \u003c 1 \\ 2x - 1, & x \geq 1 \end{cases}. Is$$ $$f(x)$$ 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 function $$f(x)$$ has a jump discontinuity at $$x = 2. $$Which of the following best describes this situation?

Accepted Answer

The left and right limits as $$x 	o 2$$ are not equal

Question 3

Consider the function $$f(x) = \frac{2x^2$ + 3x - 5}{x + 5}. At$$ $$x = -5$$, which type of discontinuity does $$f(x)$$ have?

Accepted Answer

Infinite discontinuity

Question 4

Which of the following functions is continuous everywhere on $$(-\infty, \infty)$$?

Accepted Answer

$$h(x) = x^3$ - 2x^2$ + 7x + 9$$

Question 5

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x, & 0 \leq x \u003c 2 \ k, & x = 2 \end{cases}. $$For what value of $$k is$$ $$f(x)$$ continuous on $$[0,2]$$?

Accepted Answer

$$k = 2$$

Question 6

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

Accepted Answer

$$f(x) = \frac{x^2$ - 1}{x - 1} at$$ $$x = 1$$

Question 7

The Intermediate Value Theorem (IVT) guarantees that a function $$f(x)$$ has a root in $$[a, b] if$$:

Accepted Answer

$$f(x) is $$continuous on $$[a, b]$$ and $$f(a)f(b) \u003c 0$$

Question 8

Let $$f(x) = \frac{1}{x^2$ - 9}. At $$which $$x-$$values is $$f(x)$$ discontinuous?

Accepted Answer

$$x = 3$$ and $$x = -3$$

Question 9

A function $$f(x) is $$defined as $$f(x) = \sin(x)$$ for $$x \in [0, \pi]. $$Which of the following statements is true?

Accepted Answer

$$f(x) is $$continuous on $$[0, \pi]$$

Question 10

Suppose $$f(x) is $$continuous on $$[0,1]$$, $$f(0) = 3$$, and $$f(1) = -2. $$According to the Intermediate Value Theorem, which of the following must be true?

Accepted Answer

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

Question 11

Which of the following best describes an infinite discontinuity?

Accepted Answer

The function approaches $$\infty or$$ $$-\infty as$$ $$x$$ approaches a point

Question 12

Let $$f(x) = \frac{x^2$ - 4}{x - 2}. $$What type of discontinuity does $$f(x)$$ have at $$x = 2$$?

Accepted Answer

Removable discontinuity

Question 13

Which of the following is a necessary condition for applying the Intermediate Value Theorem to $$f(x) on$$ $$[a, b]$$?

Accepted Answer

$$f(x) is $$continuous on $$[a, b]$$

Question 14

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$, & x \u003c 1 \ 2x - 1, & x \geq 1 \end{cases}. Is$$ $$f(x)$$ 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 15

A function $$f(x) is $$defined as $$f(x) = \sin\left(\frac{1}{x}ight)$$ for $$x 
eq 0$$ and $$f(0) = 0. $$What type of discontinuity does $$f(x)$$ have at $$x = 0$$?

Accepted Answer

Oscillating discontinuity

Question 16

Which of the following real-world scenarios best illustrates the Intermediate Value Theorem?

Accepted Answer

A car's speedometer reads 30 mph at 8:00 am and 50 mph at 8:10 am, so at some time between 8:00 and 8:10, the car was traveling exactly 40 mph.

Question 17

Let $$f(x) = \frac{x^2$ - 9}{x - 3}. $$What is the value of $$\lim\limits_{x 	o 3} f(x)$$?

Accepted Answer

6

Question 18

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

Accepted Answer

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

Question 19

Which of the following statements about continuity is FALSE?

Accepted Answer

A function can be continuous at a point even if it is not defined there.

Continuity and Types of Discontinuities in Calculus

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