Q: Which of the following best describes a removable discontinuity at $$x=a$$ for a function $$f(x)$$?

The limit $$\lim_{x \to a} f(x)$$ exists, but $$f(a) \neq \lim_{x \to a} f(x).$$

Q: A function $$f is $$continuous from the right at $$a if$$:

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

Q: Let $$f(x) = \frac{x^2$ - 4}{x - 2}$$ for $$x \neq 2$$ and $$f(2) = 4. Is$$ $$f(x)$$ continuous at $$x=2$$?

Yes, because $$f(2) = \lim_{x \to 2} f(x)$$

Q: A function $$f(x)$$ has an infinite discontinuity at $$x=a if$$:

$$\lim_{x \to a} f(x) = \pm \infty$$

Q: If $$f(x)$$ has a hole in its graph at $$x=a$$, which of the following is true?

$$f(a) is $$not defined, but $$\lim_{x \to a} f(x)$$ exists$

Question 1

Which of the following best describes a removable discontinuity at $$x=a$$ for a function $$f(x)$$?

Accepted Answer

The limit $$\lim_{x 	o a} f(x)$$ exists, but $$f(a) 
eq \lim_{x 	o a} f(x).$$

Question 2

Given $$f(x) = \frac{x^2$ - x - 2}{x - 2}$$ for $$x 
eq 2$$ and $$f(2) = 1$$, what type of discontinuity does $$f(x)$$ have at $$x=2$$?

Accepted Answer

Removable discontinuity

Question 3

For the function $$f(x) = [x] ($$the greatest integer function), what type of discontinuity occurs at $$x=1$$?

Accepted Answer

Jump discontinuity

Question 4

Suppose $$f(x) is $$defined as $$f(x) = \frac{1}{x-4}. $$What happens at $$x=4$$?

Accepted Answer

Infinite discontinuity

Question 5

Which of the following is TRUE about the Intermediate Value Theorem?

Accepted Answer

It guarantees a root for $$f(x) in$$ $$[a, b] if$$ $$f(a)$$ and $$f(b)$$ have opposite signs.

Question 6

Let $$f(x) = \begin{cases} x^2$ - 4 & x \u003c 2 \ ax^2$ - bx + 3 & x \geq 2 \end{cases}. $$What condition must $$a$$ and $$b$$ satisfy for $$f(x) to be $$continuous at $$x=2$$?

Accepted Answer

$$4a - 2b + 3 = 4$$

Question 7

A function $$f is $$continuous from the right at $$a if$$:

Accepted Answer

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

Question 8

Consider $$f(x) = \begin{cases} e^x & x \leq 0 \ 2 - x & x \u003e 0 \end{cases}. At$$ $$x=0$$, is $$f$$ continuous, and if not, what type of discontinuity is present?

Accepted Answer

Jump discontinuity

Question 9

If $$\lim_{x 	o a^-} f(x) = L$$ and $$\lim_{x 	o a^+} f(x) = M$$ with $$L 
eq M$$, what type of discontinuity does $$f(x)$$ have at $$x=a$$?

Accepted Answer

Jump discontinuity

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

Let $$f(x) = \frac{x^2$ - 4}{x - 2}$$ for $$x 
eq 2$$ and $$f(2) = 4. Is$$ $$f(x)$$ continuous at $$x=2$$?

Accepted Answer

Yes, because $$f(2) = \lim_{x 	o 2} f(x)$$

Question 13

A function $$f(x)$$ has an infinite discontinuity at $$x=a if$$:

Accepted Answer

$$\lim_{x 	o a} f(x) = \pm \infty$$

Question 14

If $$f(x) is $$only continuous from the right at $$a$$ and only from the left at $$b$$, which statement is true?

Accepted Answer

$$f(x) is $$not continuous at $$a$$ and $$b$$

Question 15

Which of the following functions is continuous everywhere?

Accepted Answer

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

Question 16

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ & x \u003c 1 \ 2x - 1 & x \geq 1 \end{cases}. $$What must be true for $$f(x) to be $$continuous at $$x=1$$?

Accepted Answer

$$1^2 = 2$$ \cdot 1 - 1$$

Question 17

If $$f(x)$$ has a hole in its graph at $$x=a$$, which of the following is true?

Accepted Answer

$$f(a) is $$not defined, but $$\lim_{x 	o a} f(x)$$ exists$

Question 18

A function $$f(x) is $$continuous on an interval $$[a, b] if$$:

Accepted Answer

$$f(x) is $$continuous at every point in $$(a, b)$$ and from the right at $$a$$ and from the left at $$b$$

Question 19

Which of the following is a real-world application of the Intermediate Value Theorem?

Accepted Answer

Guaranteeing that a chemical reaction reaches a certain temperature between two measured values

Continuity and Types of Discontinuity in Functions

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