Continuity of piecewise functions worksheet

A piecewise function is continuous at a break when the two pieces arrive at the same height. Write that as an equation at every break, one equation for every unknown, and solve. Differentiable asks for one more equation at each break: the slopes have to match too.

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Sigma Prep · sigmaprep.io/worksheets Name Continuity of Piecewise Functions Date Period Find the values that make each function continuous, or as asked. Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} k x + 5, & x < π \\ \cos x, & x \geq π \end{cases}$ Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} x^{2} + 16, & x < - 2 \\ a x^{2} + b, & - 2 \leq x < 1 \\ - x + 12, & x \geq 1 \end{cases}$ Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq - 1 \\ - x^{2} + 2 x - 6, & x > - 1 \end{cases}$ Find $k$ so that $f$ is continuous everywhere: $f (x) =$ ${\begin{cases} \frac{e^{4 x} - 1}{x}, & x \neq 0 \\ k, & x = 0 \end{cases}$ Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} + 2, & x < - 1 \\ a x^{2} + b, & - 1 \leq x < 3 \\ x - 18, & x \geq 3 \end{cases}$ Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq 2 \\ 16 x - 20, & x > 2 \end{cases}$ Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} k \cos x, & x < 0 \\ - 2 x - 2, & x \geq 0 \end{cases}$ Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} + 7, & x < 0 \\ a x^{2} + b, & 0 \leq x < 4 \\ x + 67, & x \geq 4 \end{cases}$ Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq - 3 \\ 24 x + 36, & x > - 3 \end{cases}$ Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} k \cos x, & x < 0 \\ 2 x + 6, & x \geq 0 \end{cases}$ Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} + 21, & x < - 3 \\ a x^{2} + b, & - 3 \leq x < - 1 \\ 2 x - 2, & x \geq - 1 \end{cases}$ Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq - 2 \\ - x^{2} - 8 x - 3, & x > - 2 \end{cases}$ Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} k \cos x, & x < 0 \\ - 4 x + 8, & x \geq 0 \end{cases}$ Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} + 1, & x < 1 \\ a x^{2} + b, & 1 \leq x < 3 \\ - x - 37, & x \geq 3 \end{cases}$ Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq - 1 \\ - x^{2} + 6 x + 11, & x > - 1 \end{cases}$ Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} x^{2} + 8, & x < 1 \\ \ln x + k, & x \geq 1 \end{cases}$ Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} - 24, & x < - 3 \\ a x^{2} + b, & - 3 \leq x < - 1 \\ 3 x + 2, & x \geq - 1 \end{cases}$ Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq 2 \\ - 4 x - 3, & x > 2 \end{cases}$ Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} x^{2} - 6, & x < 1 \\ \ln x + k, & x \geq 1 \end{cases}$ Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} x^{2} + 1, & x < 1 \\ a x^{2} + b, & 1 \leq x < 5 \\ 4 x - 42, & x \geq 5 \end{cases}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Continuity of Piecewise Functions

Date Period

Find the values that make each function continuous, or as asked.

Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} k x + 5, & x < π \\ \cos x, & x \geq π \end{cases}$
Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} x^{2} + 16, & x < - 2 \\ a x^{2} + b, & - 2 \leq x < 1 \\ - x + 12, & x \geq 1 \end{cases}$
Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq - 1 \\ - x^{2} + 2 x - 6, & x > - 1 \end{cases}$
Find $k$ so that $f$ is continuous everywhere: $f (x) =$ ${\begin{cases} \frac{e^{4 x} - 1}{x}, & x \neq 0 \\ k, & x = 0 \end{cases}$
Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} + 2, & x < - 1 \\ a x^{2} + b, & - 1 \leq x < 3 \\ x - 18, & x \geq 3 \end{cases}$
Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq 2 \\ 16 x - 20, & x > 2 \end{cases}$
Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} k \cos x, & x < 0 \\ - 2 x - 2, & x \geq 0 \end{cases}$
Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} + 7, & x < 0 \\ a x^{2} + b, & 0 \leq x < 4 \\ x + 67, & x \geq 4 \end{cases}$
Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq - 3 \\ 24 x + 36, & x > - 3 \end{cases}$
Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} k \cos x, & x < 0 \\ 2 x + 6, & x \geq 0 \end{cases}$
Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} + 21, & x < - 3 \\ a x^{2} + b, & - 3 \leq x < - 1 \\ 2 x - 2, & x \geq - 1 \end{cases}$
Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq - 2 \\ - x^{2} - 8 x - 3, & x > - 2 \end{cases}$
Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} k \cos x, & x < 0 \\ - 4 x + 8, & x \geq 0 \end{cases}$
Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} + 1, & x < 1 \\ a x^{2} + b, & 1 \leq x < 3 \\ - x - 37, & x \geq 3 \end{cases}$
Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq - 1 \\ - x^{2} + 6 x + 11, & x > - 1 \end{cases}$
Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} x^{2} + 8, & x < 1 \\ \ln x + k, & x \geq 1 \end{cases}$
Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} - 24, & x < - 3 \\ a x^{2} + b, & - 3 \leq x < - 1 \\ 3 x + 2, & x \geq - 1 \end{cases}$
Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq 2 \\ - 4 x - 3, & x > 2 \end{cases}$
Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} x^{2} - 6, & x < 1 \\ \ln x + k, & x \geq 1 \end{cases}$
Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} x^{2} + 1, & x < 1 \\ a x^{2} + b, & 1 \leq x < 5 \\ 4 x - 42, & x \geq 5 \end{cases}$



Sigma Prep · sigmaprep.io/worksheets Name Continuity of Piecewise FunctionsAnswer keyVersion 1 Date Period Find the values that make each function continuous, or as asked. Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} k x + 5, & x < π \\ \cos x, & x \geq π \end{cases}$ $k = - \frac{6}{π}$ Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} x^{2} + 16, & x < - 2 \\ a x^{2} + b, & - 2 \leq x < 1 \\ - x + 12, & x \geq 1 \end{cases}$ $a = 3, b = 8$ Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq - 1 \\ - x^{2} + 2 x - 6, & x > - 1 \end{cases}$ $a = - 2, b = - 7$ Find $k$ so that $f$ is continuous everywhere: $f (x) =$ ${\begin{cases} \frac{e^{4 x} - 1}{x}, & x \neq 0 \\ k, & x = 0 \end{cases}$ $k = 4$ Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} + 2, & x < - 1 \\ a x^{2} + b, & - 1 \leq x < 3 \\ x - 18, & x \geq 3 \end{cases}$ $a = - 2, b = 3$ Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq 2 \\ 16 x - 20, & x > 2 \end{cases}$ $a = 4, b = - 4$ Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} k \cos x, & x < 0 \\ - 2 x - 2, & x \geq 0 \end{cases}$ $k = - 2$ Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} + 7, & x < 0 \\ a x^{2} + b, & 0 \leq x < 4 \\ x + 67, & x \geq 4 \end{cases}$ $a = 4, b = 7$ Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq - 3 \\ 24 x + 36, & x > - 3 \end{cases}$ $a = - 4, b = 0$ Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} k \cos x, & x < 0 \\ 2 x + 6, & x \geq 0 \end{cases}$ $k = 6$ Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} + 21, & x < - 3 \\ a x^{2} + b, & - 3 \leq x < - 1 \\ 2 x - 2, & x \geq - 1 \end{cases}$ $a = 2, b = - 6$ Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq - 2 \\ - x^{2} - 8 x - 3, & x > - 2 \end{cases}$ $a = 1, b = 5$ Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} k \cos x, & x < 0 \\ - 4 x + 8, & x \geq 0 \end{cases}$ $k = 8$ Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} + 1, & x < 1 \\ a x^{2} + b, & 1 \leq x < 3 \\ - x - 37, & x \geq 3 \end{cases}$ $a = - 5, b = 5$ Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq - 1 \\ - x^{2} + 6 x + 11, & x > - 1 \end{cases}$ $a = - 4, b = 8$ Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} x^{2} + 8, & x < 1 \\ \ln x + k, & x \geq 1 \end{cases}$ $k = 9$ Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} - 24, & x < - 3 \\ a x^{2} + b, & - 3 \leq x < - 1 \\ 3 x + 2, & x \geq - 1 \end{cases}$ $a = - 4, b = 3$ Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq 2 \\ - 4 x - 3, & x > 2 \end{cases}$ $a = - 1, b = - 7$ Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} x^{2} - 6, & x < 1 \\ \ln x + k, & x \geq 1 \end{cases}$ $k = - 5$ Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} x^{2} + 1, & x < 1 \\ a x^{2} + b, & 1 \leq x < 5 \\ 4 x - 42, & x \geq 5 \end{cases}$ $a = - 1, b = 3$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Continuity of Piecewise FunctionsAnswer keyVersion 1

Date Period

Find the values that make each function continuous, or as asked.

Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} k x + 5, & x < π \\ \cos x, & x \geq π \end{cases}$ $k = - \frac{6}{π}$
Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} x^{2} + 16, & x < - 2 \\ a x^{2} + b, & - 2 \leq x < 1 \\ - x + 12, & x \geq 1 \end{cases}$ $a = 3, b = 8$
Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq - 1 \\ - x^{2} + 2 x - 6, & x > - 1 \end{cases}$ $a = - 2, b = - 7$
Find $k$ so that $f$ is continuous everywhere: $f (x) =$ ${\begin{cases} \frac{e^{4 x} - 1}{x}, & x \neq 0 \\ k, & x = 0 \end{cases}$ $k = 4$
Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} + 2, & x < - 1 \\ a x^{2} + b, & - 1 \leq x < 3 \\ x - 18, & x \geq 3 \end{cases}$ $a = - 2, b = 3$
Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq 2 \\ 16 x - 20, & x > 2 \end{cases}$ $a = 4, b = - 4$
Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} k \cos x, & x < 0 \\ - 2 x - 2, & x \geq 0 \end{cases}$ $k = - 2$
Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} + 7, & x < 0 \\ a x^{2} + b, & 0 \leq x < 4 \\ x + 67, & x \geq 4 \end{cases}$ $a = 4, b = 7$
Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq - 3 \\ 24 x + 36, & x > - 3 \end{cases}$ $a = - 4, b = 0$
Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} k \cos x, & x < 0 \\ 2 x + 6, & x \geq 0 \end{cases}$ $k = 6$
Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} + 21, & x < - 3 \\ a x^{2} + b, & - 3 \leq x < - 1 \\ 2 x - 2, & x \geq - 1 \end{cases}$ $a = 2, b = - 6$
Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq - 2 \\ - x^{2} - 8 x - 3, & x > - 2 \end{cases}$ $a = 1, b = 5$
Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} k \cos x, & x < 0 \\ - 4 x + 8, & x \geq 0 \end{cases}$ $k = 8$
Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} + 1, & x < 1 \\ a x^{2} + b, & 1 \leq x < 3 \\ - x - 37, & x \geq 3 \end{cases}$ $a = - 5, b = 5$
Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq - 1 \\ - x^{2} + 6 x + 11, & x > - 1 \end{cases}$ $a = - 4, b = 8$
Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} x^{2} + 8, & x < 1 \\ \ln x + k, & x \geq 1 \end{cases}$ $k = 9$
Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - x^{2} - 24, & x < - 3 \\ a x^{2} + b, & - 3 \leq x < - 1 \\ 3 x + 2, & x \geq - 1 \end{cases}$ $a = - 4, b = 3$
Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} a x^{2} + b, & x \leq 2 \\ - 4 x - 3, & x > 2 \end{cases}$ $a = - 1, b = - 7$
Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} x^{2} - 6, & x < 1 \\ \ln x + k, & x \geq 1 \end{cases}$ $k = - 5$
Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} x^{2} + 1, & x < 1 \\ a x^{2} + b, & 1 \leq x < 5 \\ 4 x - 42, & x \geq 5 \end{cases}$ $a = - 1, b = 3$

How to do these

Find $a$ and $b$ so that $f$ is continuous: $f (x) = {\begin{cases} x + 1, & x < 1 \\ a x + b, & 1 \leq x < 3 \\ 2 x, & x \geq 3 \end{cases}$

At x = 1: the left piece gives 2, so a + b = 2.
At x = 3: the right piece gives 6, so 3a + b = 6.
Subtract: 2a = 4, so a = 2 and b = 0.

The answer is $a = 2, b = 0$ .

Where students go wrong

Matching only one break. With two unknowns there are two breaks to match, and an answer that fixes the left one usually leaves a jump at the right one.

Also called making a function continuous, find k to make the function continuous or continuity with parameters.

What you can put on this worksheet

The value that makes it continuous: Find $k$ so that $f$ is continuous: $f (x) = {\begin{cases} k x - 4 & x \leq 2 \\ 0 & x > 2 \end{cases}$ → $2$
Trig, exponential, log and removable pieces: Find $k$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} e^{x} + k, & x < 0 \\ x + 2, & x \geq 0 \end{cases}$ → $k = 1$
Two unknowns, three pieces: Find $a$ and $b$ so that $f$ is continuous everywhere: $f (x) = {\begin{cases} - 3 x + 6, & x < 1 \\ a x + b, & 1 \leq x < 4 \\ 2 x - 2, & x \geq 4 \end{cases}$ → $a = 1, b = 2$
Continuous and differentiable: find a and b: Find $a$ and $b$ so that $f$ is continuous and differentiable everywhere: $f (x) = {\begin{cases} x^{2} + 3, & x \leq 2 \\ a x + b, & x > 2 \end{cases}$ → $a = 4, b = - 1$

Questions about these worksheets

Yes. Every continuity of piecewise functions sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the continuity of piecewise functions questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 4: the value that makes it continuous, trig, exponential, log and removable pieces, two unknowns, three pieces and continuous and differentiable: find a and b. Tick as many as you want and set how many of each, or let it spread them evenly.

Matching only one break. With two unknowns there are two breaks to match, and an answer that fixes the left one usually leaves a jump at the right one.

Continuity of piecewise functions worksheet

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Continuity of Piecewise Functions

Continuity of Piecewise FunctionsAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

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