Q: A function $f(x)$ is continuous at $x = c$ if which of the following conditions are satisfied?

$f(c)$ exists, $\lim_{x \to c} f(x)$ exists, and $\lim_{x \to c} f(x) = f(c)$.

Q: Consider the function $g(x) = \begin{cases} x + 2 & x \neq 2 \\ 3 & x = 2 \end{cases}$. Which statement best describes its continuity at $x = 2$?

It is discontinuous at $x = 2$ because $\lim_{x \to 2} g(x) \neq g(2)$.

Question 1

Which of the following functions is continuous at $x = 2$?

Accepted Answer

$k(x) = x + 2$

Question 2

A function $f(x)$ is continuous at $x = c$ if which of the following conditions are satisfied?

Accepted Answer

$f(c)$ exists, $\lim_{x 	o c} f(x)$ exists, and $\lim_{x 	o c} f(x) = f(c)$.

Question 3

Consider the function $g(x) = \begin{cases} x + 2 & x 
eq 2 \ 3 & x = 2 \end{cases}$. Which statement best describes its continuity at $x = 2$?

Accepted Answer

It is discontinuous at $x = 2$ because $\lim_{x 	o 2} g(x) 
eq g(2)$.

Question 4

Which type of discontinuity is present in the function $h(x) = \frac{|x - 2|}{x - 2}$ at $x = 2$?

Accepted Answer

Jump discontinuity

Question 5

If $f(x)$ and $g(x)$ are continuous at $x = c$, which of the following functions is NOT guaranteed to be continuous at $x = c$?

Accepted Answer

$h(x) = \frac{f(x)}{g(x)}$ (if $g(c) = 0$)

Question 6

Which of the following statements about polynomial functions is TRUE?

Accepted Answer

Polynomial functions are continuous at every real number.

Question 7

A company uses a 'just in time' inventory system where inventory is restocked at the start of each month and depleted linearly. Which mathematical feature best describes the inventory function?

Accepted Answer

It is a piecewise function with jump discontinuities at each integer month.

Question 8

For the function $r(x) = \frac{1}{x - 2}$, what type of discontinuity occurs at $x = 2$?

Accepted Answer

Infinite (essential) discontinuity

Question 9

Suppose $f(x) = \sqrt{1 - x^2}$ is defined on $[-1, 1]$. Which statement about its continuity at the endpoints is correct?

Accepted Answer

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

Question 10

If $f(a) < 0$ and $f(b) > 0$ for a continuous function $f(x)$ on $[a, b]$, what can be concluded?

Accepted Answer

There exists at least one $r$ in $[a, b]$ such that $f(r) = 0$.

Question 11

Given $f(x) = x^2 - 2$, estimate $\sqrt{2}$ using the Intermediate Value Theorem between $x = 1.41$ and $x = 1.42$. Which statement is correct?

Accepted Answer

$f(1.41) < 0$ and $f(1.42) > 0$, so $\sqrt{2}$ is between $1.41$ and $1.42$.

Question 12

Which of the following is a graphical interpretation of continuity at $x = c$?

Accepted Answer

The graph does not break at $x = c$; the point $(c, f(c))$ is where the function tends and actually reaches.

Question 13

Which algebraic property of continuity allows you to conclude that $h(x) = f(x) + g(x)$ is continuous at $x = c$ if $f(x)$ and $g(x)$ are continuous at $x = c$?

Accepted Answer

Sum property of continuity

Question 14

Which of the following functions is NOT continuous at $x = 2$ due to a removable discontinuity?

Accepted Answer

$f(x) = \frac{x^2 - 4}{x - 2}$

Question 15

A function $q(x) = \begin{cases} \frac{x - 2}{|x - 2|} & x 
eq 2 \ 1/2 & x = 2 \end{cases}$ has which type of discontinuity at $x = 2$?

Accepted Answer

Jump discontinuity

Question 16

Which of the following is an example of a function with a jump discontinuity in a real-world context?

Accepted Answer

A company's inventory level that is restocked at the start of each month.

Question 17

If $f(x)$ is continuous on $(a, b)$, which of the following statements is TRUE?

Accepted Answer

There are no breaks, holes, jumps, or asymptotes in the graph of $y = f(x)$ on $(a, b)$.

Question 18

Which property of limits is used to show that the sum of two continuous functions is continuous?

Accepted Answer

Limits of sums equal the sum of the individual limits.

Continuity of Functions: Definitions, Properties, and Applications

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