Types of Continuity and Discontinuity Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Types of Continuity and Discontinuity.

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

Concept Recap

Continuity types classify how a function can fail to be continuous at a point. A removable discontinuity (hole) occurs when the limit exists but doesn't equal f(a). A jump discontinuity occurs when left and right limits differ. An infinite discontinuity occurs when the function approaches ±∞.

Continuous means you can draw the graph without lifting your pen. A removable discontinuity is a single hole you could fill in. A jump discontinuity is a gap where the function leaps to a different value. An infinite discontinuity is where the function shoots off to infinity (a vertical asymptote).

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 function fails continuity at a point as a hole (removable), a jump, or a blow-up to infinity.

Common stuck point: The procedure for types of continuity and discontinuity is the easy part; the trap is calling a removable hole a jump. Asking "At the bad point, do the one-sided limits agree (hole if value mismatches), disagree (jump), or run to infinity (infinite)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: At the bad point, do the one-sided limits agree (hole if value mismatches), disagree (jump), or run to infinity (infinite)?

Worked Examples

Example 1

easy

Classify the discontinuity of

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

x = 2

Answer

Removable discontinuity at

x = 2

(hole at the point

(2, 4)

First step

x=2

: denominator

= 0

, so

f(2)

is undefined.

Full solution

2
Simplify (for $x \neq 2$ ): $\frac{(x-2)(x+2)}{x-2} = x+2$ .
3
Limit: $\lim_{x\to 2}(x+2) = 4$ . The limit exists.
4
Since the limit exists but $f(2)$ is undefined, this is a removable discontinuity (hole at $(2,4)$ ).

A removable discontinuity occurs when the two-sided limit exists but doesn't equal the function value (or the function is undefined there). It can be 'removed' by defining

f(2) = 4

Example 2

medium

Determine whether the piecewise function

f(x) = \begin{cases} x^2 & x < 1 \\ 3 - x & x \geq 1 \end{cases}

is continuous at

x = 1

Example 3

medium

Find

c

so that

f(x)=\begin{cases} 2x+c & x\le 0\\ x^2 - 3 & x>0 \end{cases}

is continuous at

x=0

Example 4

medium

Locate and classify all discontinuities of

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

Example 5

hard

Find all

a,b

that make

f(x)=\begin{cases} ax+b & x\le 1\\ x^2 - 2 & 1<x<3\\ 4x + b & x\ge 3 \end{cases}

continuous on

\mathbb{R}

Example 6

hard

Determine

k

so that

f(x) = \begin{cases} \dfrac{\sin(3x)}{x} & x\ne 0\\ k & x = 0 \end{cases}

is continuous at

0

Practice Problems

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

Example 1

easy

Classify the discontinuity of

g(x) = \frac{1}{(x-3)^2}

x = 3

Example 2

medium

Find the value of

c

that makes

h(x) = \begin{cases} cx + 1 & x \leq 2 \\ x^2 - 1 & x > 2 \end{cases}

continuous at

x = 2

Example 3

easy

Classify the discontinuity of

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

x = 2

Example 4

easy

Classify the discontinuity of the step function

f(x)

where

f(x)=1

for

x<0

and

f(x)=2

for

x\geq0

, at

x=0

Example 5

easy

Classify the discontinuity of

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

x = 0

Example 6

easy

f(x) = x^2

continuous at

x = 3

Example 7

easy

State the three conditions for

f

to be continuous at

x = a

Example 8

easy

Classify the discontinuity of

f(x) = \frac{x - 3}{x^2 - 9}

x = 3

Example 9

easy

x=1

\lim_{x\to1^-} f = 4

\lim_{x\to1^+} f = 4

, but

f(1) = 7

. Classify.

Example 10

easy

f(x) = |x|

continuous at

x = 0

Example 11

medium

Find the value of

k

that makes

f(x) = \begin{cases} x+1 & x<2 \\ kx & x\geq2 \end{cases}

continuous at

x=2

Example 12

medium

Classify the discontinuity of

f(x) = \frac{\sin x}{x}

x = 0

Example 13

medium

Where is

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

discontinuous, and of what type?

Example 14

medium

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

continuous at

x=1

Example 15

medium

Find

a

and

b

f(x)=\begin{cases} x^2 & x<1 \\ ax+b & 1\leq x \leq 3 \\ 12 & x>3 \end{cases}

is continuous everywhere.

Example 16

medium

Classify the discontinuity of

f(x) = \frac{|x|}{x}

x = 0

Example 17

medium

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

, with

f(1)

defined as

2

, continuous at

x = 1

Example 18

medium

x=0

\lim_{x\to0^-} f = 2

and

\lim_{x\to0^+} f = +\infty

. Classify.

Example 19

challenge

For what value of

c

does

f(x) = \frac{x^2 + cx - 6}{x - 2}

have a removable discontinuity at

x = 2

Example 20

medium

Determine all discontinuities of

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

and classify each.

Example 21

challenge

f(x) = x \sin\!\left(\frac{1}{x}\right)

for

x \neq 0

f(0)=0

, continuous at 0?

Example 22

challenge

Show

f(x) = \begin{cases} \frac{e^{2x}-1}{x} & x\neq0 \\ k & x=0 \end{cases}

is continuous at 0; find

k

Example 23

easy

Classify the discontinuity of

f(x) = \dfrac{x^2 - 9}{x - 3}

x = 3

Example 24

easy

Classify the discontinuity of

f(x) = \dfrac{1}{x - 5}

x = 5

Example 25

easy

f(x) = \lfloor x \rfloor

continuous at

x = 2

Example 26

easy

Classify the discontinuity of

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

x = -2

Example 27

easy

Classify the discontinuity of

f(x) = \tan x

x = \dfrac{\pi}{2}

Example 28

medium

Classify the discontinuity of

f(x)=\dfrac{x^2 - 5x + 6}{x - 2}

x = 2

Example 29

medium

Find

a

so that

f(x)=\begin{cases} a x^2 & x\le 1\\ 4x - 2 & x>1 \end{cases}

is continuous at

x=1

Example 30

medium

Classify the discontinuity at

x=0

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

Example 31

medium

Classify the discontinuity of

f(x) = e^{1/x}

x = 0

Example 32

medium

For what value of

k

f(x)=\begin{cases}\dfrac{x^2-16}{x-4} & x\ne 4\\ k & x=4\end{cases}

continuous at

x=4

Example 33

medium

Find

a,b

f(x)=\begin{cases} x+a & x<0\\ b & x=0\\ x^2+1 & x>0 \end{cases}

is continuous at

0

Example 34

hard

Classify the discontinuity at

x = 0

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

Example 35

hard

Classify the discontinuity at

x = 0

f(x) = \sin\!\left(\dfrac{1}{x}\right)

Example 36

hard

Find

c

so that

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

has a removable discontinuity at

x=1

Example 37

medium

Classify the discontinuity of

f(x) = \dfrac{x^3 - 8}{x - 2}

x = 2

Example 38

medium

At which

x

values is

f(x) = \dfrac{x^2}{x^2 - 5x + 6}

discontinuous, and of what type?

Example 39

challenge

Show that

f(x) = \begin{cases} x & x\in\mathbb{Q}\\ -x & x\notin\mathbb{Q} \end{cases}

is continuous only at

x = 0

Example 40

challenge

Find all

a

such that

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

has a removable discontinuity at

x = a

for every real

a

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Related Concepts

Limit Intermediate Value Theorem Squeeze Theorem

Background Knowledge

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

limit