Worksheets · Calculus

Continuity and discontinuities worksheet for Calculus

A rational function breaks wherever its bottom is zero. Whether that is a hole you could fill in or an asymptote it runs off along depends on whether the factor cancels. The sheets also ask for the constant that joins two pieces, whether the Intermediate Value Theorem guarantees a zero, and where a function is continuous.

Also taught in Precalculus, at its own level and with its own examples.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

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Name

Continuity and Discontinuities

Date Period

Answer each question about the function.

  1. Classify a discontinuity

    Classify the discontinuity of f(x)=(x−2)(x−1)(x−2)(x+4) at x=−4
  2. Classify the discontinuity of f(x)=(x−3)(x+1)(x−3)(x+3) at x=−3
  3. Classify the discontinuity of f(x)=(x+4)(x+3)(x+4)(x−3) at x=3
  4. Classify the discontinuity of f(x)=(x−1)(x+4)(x−1)(x+2) at x=1

  1. Find where a function is discontinuous

    Where is f(x)=(x−4)(x−2)(x−4)(x+2) discontinuous?
  2. Where is f(x)=(x+3)(x+1)(x+3)(x+2) discontinuous?
  3. Where is f(x)=(x−1)(x−3)(x−1)(x−4) discontinuous?
  4. Where is f(x)=(x−3)(x+5)(x−3)(x+4) discontinuous?

  1. Find kk that makes ff continuous

    Find k so that f is continuous: f(x)={kx−2x≤23xx>2
  2. Find k so that f is continuous: f(x)={kx−9x≤−13x−7x>−1
  3. Find k so that f is continuous: f(x)={kxx≤−3x+9x>−3
  4. Find k so that f is continuous: f(x)={kx+8x≤25x−8x>2

  1. Decide what the Intermediate Value Theorem guarantees

    Does the Intermediate Value Theorem guarantee that f(x)=2x3−3x−6 has a zero on [−3,−2]?
  2. Does the Intermediate Value Theorem guarantee that f(x)=−x3+5x−5 has a zero on [2,4]?
  3. Does the Intermediate Value Theorem guarantee that f(x)=x3−x−3 has a zero on [0,2]?
  4. Does the Intermediate Value Theorem guarantee that f(x)=2x3+4x−5 has a zero on [−3,−2]?

  1. Find the intervals of continuity

    Give the intervals on which f(x)=3x2−2x+8 is continuous.
  2. Give the intervals on which f(x)=x+5 is continuous.
  3. Give the intervals on which f(x)=3xx2+2x−8 is continuous.
  4. Give the intervals on which f(x)=−xx2+2x−15 is continuous.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Continuity and DiscontinuitiesAnswer keyVersion 1

Date Period

Answer each question about the function.

  1. Classify a discontinuity

    Classify the discontinuity of f(x)=(x−2)(x−1)(x−2)(x+4) at x=−4Infinite
  2. Classify the discontinuity of f(x)=(x−3)(x+1)(x−3)(x+3) at x=−3Infinite
  3. Classify the discontinuity of f(x)=(x+4)(x+3)(x+4)(x−3) at x=3Infinite
  4. Classify the discontinuity of f(x)=(x−1)(x+4)(x−1)(x+2) at x=1Removable

  1. Find where a function is discontinuous

    Where is f(x)=(x−4)(x−2)(x−4)(x+2) discontinuous?x=−2, x=4
  2. Where is f(x)=(x+3)(x+1)(x+3)(x+2) discontinuous?x=−3, x=−2
  3. Where is f(x)=(x−1)(x−3)(x−1)(x−4) discontinuous?x=1, x=4
  4. Where is f(x)=(x−3)(x+5)(x−3)(x+4) discontinuous?x=−4, x=3

  1. Find kk that makes ff continuous

    Find k so that f is continuous: f(x)={kx−2x≤23xx>24
  2. Find k so that f is continuous: f(x)={kx−9x≤−13x−7x>−11
  3. Find k so that f is continuous: f(x)={kxx≤−3x+9x>−3−2
  4. Find k so that f is continuous: f(x)={kx+8x≤25x−8x>2−3

  1. Decide what the Intermediate Value Theorem guarantees

    Does the Intermediate Value Theorem guarantee that f(x)=2x3−3x−6 has a zero on [−3,−2]?No: f(−3)=−51 and f(−2)=−16 have the same sign
  2. Does the Intermediate Value Theorem guarantee that f(x)=−x3+5x−5 has a zero on [2,4]?No: f(2)=−3 and f(4)=−49 have the same sign
  3. Does the Intermediate Value Theorem guarantee that f(x)=x3−x−3 has a zero on [0,2]?Yes: f(0)=−3 and f(2)=3
  4. Does the Intermediate Value Theorem guarantee that f(x)=2x3+4x−5 has a zero on [−3,−2]?No: f(−3)=−71 and f(−2)=−29 have the same sign

  1. Find the intervals of continuity

    Give the intervals on which f(x)=3x2−2x+8 is continuous.(−∞,∞)
  2. Give the intervals on which f(x)=x+5 is continuous.[−5,∞)
  3. Give the intervals on which f(x)=3xx2+2x−8 is continuous.(−∞,−4)∪(−4,2)∪(2,∞)
  4. Give the intervals on which f(x)=−xx2+2x−15 is continuous.(−∞,−5)∪(−5,3)∪(3,∞)

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Classify the discontinuity of f(x)=(x−2)(x−2)(x+3) at x=2

  1. The factor x minus 2 is on both the top and the bottom.
  2. So it cancels, and the limit there is finite.
  3. That makes it a removable discontinuity, a hole.

The answer is Removable.

Where students go wrong

Calling every zero of the bottom an asymptote. If the same factor is on the top it cancels, and what is left is a single missing point rather than a curve running off to infinity.

Also called removable discontinuity, infinite discontinuity, holes and asymptotes, determining continuity or classifying discontinuities.

What you can put on this worksheet

Classify a discontinuity
Classify the discontinuity of f(x)=(x−1)(x−1)(x−2) at x=1 → Removable
Where it is discontinuous
Where is f(x)=(x−1)(x−1)(x−2) discontinuous? → x=1, x=2
Piecewise functions: continuous, jump, removable or infinite
Is f continuous at x=3? If not, name the discontinuity.f(x)={−3x+1x≤3−5x+2x>3 → Jump discontinuity
The value that makes it continuous
Find k so that f is continuous: f(x)={kx−4x≤20x>2 → 2
The Intermediate Value Theorem
Does the Intermediate Value Theorem guarantee that f(x)=−x3+x+4 has a zero on [−3,−2]? → No: f(−3)=28 and f(−2)=10 have the same sign
Intervals of continuity
Give the intervals on which f(x)=x+7x+3 is continuous. → (−∞,−3)∪(−3,∞)

Questions about these worksheets

Yes. Every continuity and discontinuities 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 and discontinuities 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 6: classify a discontinuity, where it is discontinuous, piecewise functions: continuous, jump, removable or infinite, the value that makes it continuous, the intermediate value theorem and intervals of continuity. Tick as many as you want and set how many of each, or let it spread them evenly.

Calling every zero of the bottom an asymptote. If the same factor is on the top it cancels, and what is left is a single missing point rather than a curve running off to infinity.