AP Calculus AB Continuity and the Intermediate Value Theorem Worksheet (PDF)

AP Calculus AB math worksheet: Continuity and the Intermediate Value Theorem – Maths PDF

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This continuity practice worksheet for AP Calculus AB covers continuity and the intermediate value theorem (Common Core AP CALC AB 1.10, AP CALC AB 1.11, AP CALC AB 1.12, AP CALC AB 1.16), with a free PDF to print.

These 21 problems practice continuity, types of discontinuity, making piecewise functions continuous and the Intermediate Value Theorem (AP Calculus AB, topics 1.10, 1.11, 1.12, 1.16). The early problems build the basic skill, and each one uses a different function or situation so that no two feel alike.

Later problems use piecewise functions with unknown constants, proving that equations have solutions, and situations such as shipping charges and a hiking trail. Work in pencil, show every step, and check the answer key only after a full attempt.

AP Calculus AB · Common Core State Standards: AP CALC AB 1.10, AP CALC AB 1.11, AP CALC AB 1.12, AP CALC AB 1.16

Continuity practice worksheet: exercises 1 to 10

Exercise 1 – Continuity of a polynomial

Is $f(x)=x^{2}+3x$ continuous at $x=1$? Check the three conditions.

Exercise 2 – Where is it undefined?

Find every $x$-value where $g(x)=\frac{x+1}{x-4}$ is discontinuous, and classify the discontinuity.

Exercise 3 – Removable discontinuity

Classify the discontinuity of $h(x)=\frac{x^{2}-9}{x-3}$ at $x=3$ and say how to define $h(3)$ to make $h$ continuous.

Exercise 4 – Types from a graph

The graph of f has trouble at x = 1 and x = 3.

Identify each discontinuity of $f$ in the graph and classify it as removable, jump or infinite.

Exercise 5 – Checking conditions from data

A function has $f(4)=7$, $\lim_{x\to 4^-} f(x)=7$ and $\lim_{x\to 4^+} f(x)=9$. Is $f$ continuous at $x=4$? Which condition fails?

Exercise 6 – Repairing a hole

The repaired graph passes through (2, 5).

Let $f(x)=\frac{x^{2}+x-6}{x-2}$ for $x\neq 2$ and $f(2)=k$. Find $k$ so that $f$ is continuous at $x=2$.

Exercise 7 – Piecewise, one unknown

Let $f(x)=x^{2}$ for $x < 1$ and $f(x)=ax+3$ for $x\ge 1$. Find $a$ so that $f$ is continuous at $x=1$.

Exercise 8 – Piecewise, a second example

Let $f(x)=2x+b$ for $x\le 3$ and $f(x)=x^{2}-1$ for $x > 3$. Find $b$ for continuity at $x=3$.

Exercise 9 – Two points to check

For $p(x)=\frac{x^{2}-4x}{x^{2}-16}$, classify the discontinuities at $x=4$ and $x=-4$.

Exercise 10 – Domain and continuity

On what interval is $f(x)=\sqrt{x-2}$ continuous? Is it continuous at $x=2$?

Exercises 11 and up

Exercise 11 – Sine over x

Define $g(x)=\frac{\sin x}{x}$ for $x\neq 0$. What value of $g(0)$ makes $g$ continuous at $0$? What if $g(0)=0$?

Exercise 12 – IVT and a cubic

f(1) is negative and f(2) is positive.

Show that $f(x)=x^{3}+x-5$ has a zero in $(1,2)$.

Exercise 13 – Two zeros

Show that $f(x)=x^{3}-4x+1$ has a zero in $(0,1)$ and another in $(1,3)$.

Exercise 14 – Which values are guaranteed?

A function $f$ is continuous on $[1,5]$ with $f(1)=-2$ and $f(5)=10$. Does the IVT guarantee a $c$ with $f(c)=4$? With $f(c)=12$?

Exercise 15 – When the IVT fails

The graph of 1/x jumps across x = 0.

For $f(x)=\frac{1}{x}$, $f(-1)=-1$ and $f(1)=1$, yet $f(x)=0$ has no solution. Why does this not contradict the IVT?

Exercise 16 – Several discontinuities

Find and classify every discontinuity of $r(x)=\frac{x^{2}-1}{x^{2}-5x+6}$.

Exercise 17 – Shipping charges

Sofia’s shop charges \$4 to ship a package up to 1 pound, \$6 for over 1 up to 2 pounds, and \$8 for over 2 up to 3 pounds. Where is the shipping cost $S(w)$ discontinuous for $0 < w\le 3$? Classify and give the size of each jump.

Exercise 18 – Hiking elevation

Mei hikes a trail where her elevation is a continuous function of time. She starts at 800 feet and finishes at 2,300 feet. Must she have been at exactly 1,500 feet at some time? Exactly 2,500 feet?

Exercise 19 – Two unknowns

Let $f(x)=2$ for $x\le 1$, $f(x)=ax+b$ for $1 < x < 3$ and $f(x)=6$ for $x\ge 3$. Find $a$ and $b$ so that $f$ is continuous everywhere.

Exercise 20 – Solving cos x = x

The curves y = cos x and y = x cross once on [0, 1].

Show that the equation $\cos x=x$ has a solution in $[0,1]$ (radians).

Exercise 21 – Solving an exponential equation

Show that $e^{x}+x=3$ has a solution in $(0,1)$.

Key facts behind this continuity practice worksheet

Key facts

Continuity needs three things: f(a) exists, the limit exists, and they are equal.
Removable discontinuities are holes; jump discontinuities have different one-sided limits; infinite discontinuities are vertical asymptotes.
To make a piecewise function continuous, set the one-sided limits equal at the boundary.
The IVT guarantees a value, but only if the function is continuous on the whole closed interval.

How to use this continuity practice worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the continuity practice worksheet as a PDF with the button below and keep it with your notes.

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