Practice Continuous Function in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

A function is continuous at a point if the limit equals the function value there, with no jumps, holes, or vertical asymptotes in the interval of interest.

A continuous function can be drawn without lifting the pencil — there are no sudden jumps, gaps, or points that shoot to infinity.

Showing a random 20 of 50 problems.

Example 1

challenge

True or false: if

f

is continuous on

[0,2]

with

f(0)=1

and

f(2)=3

, then

f

must have a fixed point in

[0,2]

Example 2

easy

Is a jump in a graph a sign of discontinuity?

Example 3

easy

Can a continuous function have a vertical asymptote within its domain?

Example 4

medium

Find

k

so that

f(x)=\begin{cases}\dfrac{x^2-25}{x-5} & x\ne 5\\ k & x=5\end{cases}

is continuous at

x=5

Example 5

hard

Show that

f(x) = \cos x - x

has a root in

[0,1]

Example 6

easy

Are removable discontinuities always visible at normal zoom?

Example 7

easy

f(x) = 5x - 7

continuous on

\mathbb{R}

Example 8

challenge

Find

k

so that

f(x)=\begin{cases}\frac{x^2-9}{x-3} & x\ne 3\\ k & x=3\end{cases}

is continuous at

x=3

Example 9

hard

f

is continuous on

[0,1]

and

f(x)\in[0,1]

for all

x\in[0,1]

, show

f

has a fixed point.

Example 10

medium

State the three conditions for

f

to be continuous at

x=a

Example 11

hard

Find

a,b

so that

f(x)=\begin{cases} ax+1 & x<2\\ x^2+b & x\ge 2 \end{cases}

is continuous at

x=2

and

f(2)=7

Example 12

easy

f(x) = \sin x

continuous on

\mathbb{R}

Example 13

medium

State the largest interval on which

f(x)=\sqrt{x-3}

is continuous.

Example 14

challenge

By the Intermediate Value Theorem, must

f(x)=x^3+x-1

have a root in

[0,1]

? Justify.

Example 15

easy

Show that

f(x) = 3x^2 - 5x + 2

is continuous at

x = 1

using the three-part definition of continuity.

Example 16

easy

f(x) = e^x

continuous on

\mathbb{R}

Example 17

medium

f

is continuous on

[0,1]

with

f(0)=2

and

f(1)=-1

, must

f

take the value

0

somewhere on

[0,1]

Example 18

medium

On what set is

f(x) = \dfrac{1}{\sin x}

continuous?

Example 19

easy

g(x) = \dfrac{1}{x}

continuous on

(-\infty, \infty)

? If not, identify where and why.

Example 20

medium

Where is

f(x)=\dfrac{x+1}{x^2-x-6}

discontinuous?

← Back to Continuous Function Worked Examples Formula Explained

Related Concepts

Function Limit Intermediate Value Theorem