Practice Continuous Function in Math

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

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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 โ€” 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 ff is continuous on [0,2][0,2] with f(0)=1f(0)=1 and f(2)=3f(2)=3, then ff must have a fixed point in [0,2][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 kk so that f(x)={x2โˆ’25xโˆ’5xโ‰ 5kx=5f(x)=\begin{cases}\dfrac{x^2-25}{x-5} & x\ne 5\\ k & x=5\end{cases} is continuous at x=5x=5.

Example 5

hard
Show that f(x)=cosโกxโˆ’xf(x) = \cos x - x has a root in [0,1][0,1].

Example 6

easy
Are removable discontinuities always visible at normal zoom?

Example 7

easy
Is f(x)=5xโˆ’7f(x) = 5x - 7 continuous on R\mathbb{R}?

Example 8

challenge
Find kk so that f(x)={x2โˆ’9xโˆ’3xโ‰ 3kx=3f(x)=\begin{cases}\frac{x^2-9}{x-3} & x\ne 3\\ k & x=3\end{cases} is continuous at x=3x=3.

Example 9

hard
If ff is continuous on [0,1][0,1] and f(x)โˆˆ[0,1]f(x)\in[0,1] for all xโˆˆ[0,1]x\in[0,1], show ff has a fixed point.

Example 10

medium
State the three conditions for ff to be continuous at x=ax=a.

Example 11

hard
Find a,ba,b so that f(x)={ax+1x<2x2+bxโ‰ฅ2f(x)=\begin{cases} ax+1 & x<2\\ x^2+b & x\ge 2 \end{cases} is continuous at x=2x=2 and f(2)=7f(2)=7.

Example 12

easy
Is f(x)=sinโกxf(x) = \sin x continuous on R\mathbb{R}?

Example 13

medium
State the largest interval on which f(x)=xโˆ’3f(x)=\sqrt{x-3} is continuous.

Example 14

challenge
By the Intermediate Value Theorem, must f(x)=x3+xโˆ’1f(x)=x^3+x-1 have a root in [0,1][0,1]? Justify.

Example 15

easy
Show that f(x)=3x2โˆ’5x+2f(x) = 3x^2 - 5x + 2 is continuous at x=1x = 1 using the three-part definition of continuity.

Example 16

easy
Is f(x)=exf(x) = e^x continuous on R\mathbb{R}?

Example 17

medium
If ff is continuous on [0,1][0,1] with f(0)=2f(0)=2 and f(1)=โˆ’1f(1)=-1, must ff take the value 00 somewhere on [0,1][0,1]?

Example 18

medium
On what set is f(x)=1sinโกxf(x) = \dfrac{1}{\sin x} continuous?

Example 19

easy
Is g(x)=1xg(x) = \dfrac{1}{x} continuous on (โˆ’โˆž,โˆž)(-\infty, \infty)? If not, identify where and why.

Example 20

medium
Where is f(x)=x+1x2โˆ’xโˆ’6f(x)=\dfrac{x+1}{x^2-x-6} discontinuous?
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