Practice Types of Continuity and Discontinuity in Math

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

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Quick 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).

Showing a random 20 of 50 problems.

Example 1

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 2

easy

Classify the discontinuity of

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

x = 3

Example 3

hard

Classify the discontinuity at

x = 0

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

Example 4

medium

Locate and classify all discontinuities of

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

Example 5

hard

Classify the discontinuity at

x = 0

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

Example 6

medium

Classify the discontinuity of

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

x = 0

Example 7

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 8

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 9

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 10

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 11

easy

Classify the discontinuity of

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

x = -2

Example 12

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 13

medium

Classify the discontinuity of

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

x = 2

Example 14

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 15

hard

Find

c

so that

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

has a removable discontinuity at

x=1

Example 16

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

Example 17

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 18

medium

x=0

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

and

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

. Classify.

Example 19

medium

List the type of discontinuity at

x=0

for

f(x) = \dfrac{\sin x}{x^2}

Example 20

easy

Classify the discontinuity of

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

x = 3

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Related Concepts

Limit Intermediate Value Theorem Squeeze Theorem