Continuous Function Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Continuous Function.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A continuous function has no jumps, holes, or breaks; its limit equals its value at every point.

Common stuck point: The procedure for continuous function is the easy part; the trap is assuming defined-everywhere means continuous. Asking "Can the graph be drawn through this point without lifting the pencil?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can the graph be drawn through this point without lifting the pencil?

Worked Examples

Example 1

easy

Show that

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

is continuous at

x = 1

using the three-part definition of continuity.

Answer

f

is continuous at

x = 1

First step

Part 1 —

f(1)

exists:

f(1) = 3(1)-5(1)+2 = 0

. ✓

Full solution

2
Part 2 — $\lim_{x\to1} f(x)$ exists: since $f$ is a polynomial, the limit equals the function value. $\lim_{x\to1}(3x^2-5x+2) = 3-5+2 = 0$ . ✓
3
Part 3 — Limit equals function value: $\lim_{x\to1}f(x) = 0 = f(1)$ . ✓ All three conditions hold, so $f$ is continuous at $x=1$ .

Continuity requires three conditions: the function value exists, the limit exists, and they are equal. Polynomials satisfy all three at every point, making them everywhere continuous.

Example 2

hard

Find where

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

is discontinuous, classify the discontinuity, and determine if it can be removed.

Example 3

medium

Apply the IVT to show

f(x)=x^3 + x - 4

has a root in

[1,2]

Example 4

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 5

hard

Show that

f(x) = \cos x - x

has a root in

[0,1]

Example 6

challenge

Show that any polynomial of odd degree has at least one real root.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

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

continuous on

(-\infty, \infty)

? If not, identify where and why.

Example 2

medium

Apply the Intermediate Value Theorem to show that

h(x) = x^3 - x - 1

has a root in the interval

[1, 2]

Example 3

easy

Can you draw

f(x)=x

without lifting your pencil?

Example 4

easy

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

continuous at

x=0

Example 5

easy

f(x)=|x|

continuous at

x=0

Example 6

easy

Does continuous mean smooth (no corners)?

Example 7

easy

Is the polynomial

f(x)=x^3-2x+1

continuous for all real

x

Example 8

easy

At a removable discontinuity (hole), is the function continuous there?

Example 9

easy

Is a jump in a graph a sign of discontinuity?

Example 10

easy

Are removable discontinuities always visible at normal zoom?

Example 11

medium

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

continuous at

x=2

Example 12

medium

For

f(x)=\begin{cases}x+2 & x<1\\ 3x & x\ge 1\end{cases}

, is

f

continuous at

x=1

Example 13

medium

Where is

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

discontinuous?

Example 14

medium

f(x)=\sqrt{x}

continuous at

x=0

Example 15

medium

A function jumps from

2

(left) to

5

(right) at

x=3

. What is the jump size?

Example 16

medium

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

continuous for all real

x

Example 17

medium

State the three conditions for

f

to be continuous at

x=a

Example 18

medium

f(x)=\tan x

continuous on

(-\tfrac{\pi}{2},\tfrac{\pi}{2})

Example 19

medium

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

continuous at

x=0

Example 20

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 21

challenge

By the Intermediate Value Theorem, must

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

have a root in

[0,1]

? Justify.

Example 22

challenge

For

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

, find a relation between

a

and

b

for continuity at

x=1

Example 23

easy

f(x) = 5x - 7

continuous on

\mathbb{R}

Example 24

easy

f(x) = e^x

continuous on

\mathbb{R}

Example 25

easy

Where is

f(x) = \dfrac{1}{x^2 - 4}

discontinuous?

Example 26

easy

f(x) = \sin x

continuous on

\mathbb{R}

Example 27

easy

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

Example 28

medium

Find the value(s) of

x

where

f(x)=\dfrac{x+3}{x^2+x-6}

is discontinuous.

Example 29

medium

Is the product of two continuous functions continuous?

Example 30

medium

For

f(x)=\begin{cases} x^2 + 1 & x\le 2\\ 3x - 1 & x > 2 \end{cases}

, is

f

continuous at

x=2

Example 31

medium

f(x) = \ln(x^2 + 1)

continuous on

\mathbb{R}

Example 32

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 33

medium

On what set is

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

continuous?

Example 34

medium

True or false: if

f

is continuous at

a

, then

|f|

is continuous at

a

Example 35

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 36

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 37

medium

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

continuous everywhere?

Example 38

medium

True or false:

f(x) = \tan x

is continuous on

[0, \pi]

Example 39

hard

Let

f

be continuous on

\mathbb{R}

with

f(x+y)=f(x)+f(y)

for all real

x,y

. Show

f(x)=cx

for some constant

c

Example 40

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]

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Related Concepts

Function Limit Intermediate Value Theorem

Background Knowledge

These ideas may be useful before you work through the harder examples.

function definition