Continuity

Name:

Class:

Date:

Show your work for each problem.

1.
Which of these is NOT one of the three conditions for $f$ to be continuous at $x = a$ ?
1. a) $\lim_{x \to a} f(x)$ exists
2. b) $f(a)$ is defined
3. c) $\lim_{x \to a} f(x) = f(a)$
4. d) $f(a) = 0$
2.
$f(x) = 3x^2 - 2x + 5$ is continuous everywhere. Find $\lim_{x \to 2} f(x)$ .
3.
Where is $f(x) = \frac{1}{x - 4}$ NOT continuous?
1. a) $x = 4$
2. b) $x = -4$
3. c) $x = 0$
4. d)It is continuous everywhere
4.
$g(x) = \frac{x^2 - 1}{x - 1}$ is undefined at $x = 1$ . Which value should $g(1)$ get so that $g$ is continuous at $x = 1$ ?
5.
Check whether $f(x) = x^2 + 2x$ is continuous at $x = 3$ .
6.
Is $f(x) = \begin{cases} 2x + 1, & x < 1 \\ 4 - x, & x \geq 1 \end{cases}$ continuous at $x = 1$ ?
7.
Which function is continuous for ALL real numbers?
1. a) $\frac{1}{x}$
2. b) $e^x + x^3$
3. c) $\sqrt{x}$
4. d) $\ln x$
8.
Find $k$ so that $f(x) = \begin{cases} x + k, & x < 2 \\ x^2, & x \geq 2 \end{cases}$ is continuous at $x = 2$ .
9.
Find $k$ so that $f(x) = \begin{cases} kx - 2, & x \leq 3 \\ x^2 + 1, & x > 3 \end{cases}$ is continuous everywhere.
10.
Let $h(x) = \frac{x^2 - 5x + 6}{x - 2}$ for $x \neq 2$ and $h(2) = 3$ . Is $h$ continuous at $x = 2$ ?
11.
Where is $f(x) = \frac{x + 3}{x^2 - 9}$ continuous?
1. a)All real numbers except $x = -3$
2. b)All real numbers except $x = 3$ and $x = -3$
3. c)All real numbers
4. d)All real numbers except $x = 3$
12.
At how many values of $x$ is $f(x) = \frac{x - 1}{x^3 - 4x}$ not continuous?
13.
Use the Intermediate Value Theorem to decide whether $f(x) = x^3 - 2x - 5$ has a root in $[2, 3]$ .
14.
Find $a$ so that $f(x) = \begin{cases} ax + 3, & x < 2 \\ x^2 + a, & x \geq 2 \end{cases}$ is continuous everywhere.
15.
Find $a$ and $b$ so that $f(x) = \begin{cases} x + 1, & x < 1 \\ ax + b, & 1 \leq x < 3 \\ x^2 - 3, & x \geq 3 \end{cases}$ is continuous everywhere.
16.
$f(x) = \frac{1}{x}$ has $f(-1) = -1$ and $f(1) = 1$ , but $f(x) = 0$ has no solution. Why does the Intermediate Value Theorem not apply on $[-1, 1]$ ?
1. a) $f(-1)$ and $f(1)$ have the same sign
2. b)The theorem only works for polynomials
3. c) $f$ is not continuous on $[-1, 1]$ : it is undefined at $x = 0$
4. d) $0$ is not between $-1$ and $1$
17.
Let $f(x) = \frac{\sqrt{x} - 2}{x - 4}$ for $x \neq 4$ and $f(4) = k$ . Find $k$ so that $f$ is continuous at $x = 4$ .
18.
How many roots of $f(x) = x^3 - 3x + 1$ in $[0, 2]$ does the Intermediate Value Theorem guarantee, using the points $x = 0$ , $1$ , $2$ ?