Worksheets · Precalculus

Precalculus continuity and discontinuities worksheet

In Precalculus continuity is checked with limits: the limit from each side has to exist, agree, and match the value of the function. That is how you find holes and asymptotes when graphing rational functions, and how you choose a constant that joins two pieces.

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

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Sigma Prep · sigmaprep.io/worksheets

Name

Continuity and Discontinuities

Date Period

Answer each question about the function.

  1. Classify the discontinuity of f(x)=(x−2)(x−1)(x−2)(x+4) at x=−4
  2. Where is f(x)=(x−4)(x−2)(x−4)(x+2) discontinuous?
  3. Find k so that f is continuous: f(x)={kx−2x≤23xx>2
  4. Does the Intermediate Value Theorem guarantee that f(x)=2x3−3x−6 has a zero on [−3,−2]?
  5. Classify the discontinuity of f(x)=(x−1)(x+3)(x−1)(x−3) at x=1
  6. Where is f(x)=(x−2)(x−3)(x−2)(x−1) discontinuous?
  7. Find k so that f is continuous: f(x)={kx+6x≤−3−4x+9x>−3
  8. Does the Intermediate Value Theorem guarantee that f(x)=x3−2x−2 has a zero on [−2,−1]?
  9. Classify the discontinuity of f(x)=(x+4)(x−4)(x+4)(x−1) at x=−4
  10. Where is f(x)=(x+2)(x−5)(x+2)(x−2) discontinuous?
  11. Find k so that f is continuous: f(x)={kx−7x≤35x+8x>3
  12. Does the Intermediate Value Theorem guarantee that f(x)=2x3−3x+7 has a zero on [0,2]?
  13. Classify the discontinuity of f(x)=(x−2)(x−5)(x−2)(x+3) at x=2
  14. Where is f(x)=(x−4)(x+2)(x−4)(x+3) discontinuous?
  15. Find k so that f is continuous: f(x)={kx−7x≤−32x−7x>−3
  16. Does the Intermediate Value Theorem guarantee that f(x)=−x3−4x+6 has a zero on [−2,0]?
  17. Classify the discontinuity of f(x)=(x+4)(x+2)(x+4)(x−4) at x=4
  18. Where is f(x)=(x+3)(x−1)(x+3)(x+1) discontinuous?
  19. Find k so that f is continuous: f(x)={kx+3x≤−47x>−4
  20. Does the Intermediate Value Theorem guarantee that f(x)=2x3+6x−6 has a zero on [−2,−1]?

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 the discontinuity of f(x)=(x−2)(x−1)(x−2)(x+4) at x=−4Infinite
  2. Where is f(x)=(x−4)(x−2)(x−4)(x+2) discontinuous?x=−2, x=4
  3. Find k so that f is continuous: f(x)={kx−2x≤23xx>24
  4. 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
  5. Classify the discontinuity of f(x)=(x−1)(x+3)(x−1)(x−3) at x=1Removable
  6. Where is f(x)=(x−2)(x−3)(x−2)(x−1) discontinuous?x=1, x=2
  7. Find k so that f is continuous: f(x)={kx+6x≤−3−4x+9x>−3−5
  8. Does the Intermediate Value Theorem guarantee that f(x)=x3−2x−2 has a zero on [−2,−1]?No: f(−2)=−6 and f(−1)=−1 have the same sign
  9. Classify the discontinuity of f(x)=(x+4)(x−4)(x+4)(x−1) at x=−4Removable
  10. Where is f(x)=(x+2)(x−5)(x+2)(x−2) discontinuous?x=−2, x=2
  11. Find k so that f is continuous: f(x)={kx−7x≤35x+8x>310
  12. Does the Intermediate Value Theorem guarantee that f(x)=2x3−3x+7 has a zero on [0,2]?No: f(0)=7 and f(2)=17 have the same sign
  13. Classify the discontinuity of f(x)=(x−2)(x−5)(x−2)(x+3) at x=2Removable
  14. Where is f(x)=(x−4)(x+2)(x−4)(x+3) discontinuous?x=−3, x=4
  15. Find k so that f is continuous: f(x)={kx−7x≤−32x−7x>−32
  16. Does the Intermediate Value Theorem guarantee that f(x)=−x3−4x+6 has a zero on [−2,0]?No: f(−2)=22 and f(0)=6 have the same sign
  17. Classify the discontinuity of f(x)=(x+4)(x+2)(x+4)(x−4) at x=4Infinite
  18. Where is f(x)=(x+3)(x−1)(x+3)(x+1) discontinuous?x=−3, x=−1
  19. Find k so that f is continuous: f(x)={kx+3x≤−47x>−4−1
  20. Does the Intermediate Value Theorem guarantee that f(x)=2x3+6x−6 has a zero on [−2,−1]?No: f(−2)=−34 and f(−1)=−14 have the same sign

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

How to do these

Find k so f(x)={x2+k,x<23x−1,x≥2 is continuous

  1. The left piece heads to 4 + k as x approaches 2.
  2. The right piece gives 3(2) - 1 = 5 at x = 2.
  3. Set them equal: 4 + k = 5.

The answer is k=1.

Where students go wrong

Checking only that f(2) exists. A function can have a value at a point and still jump there; the left and right limits must both equal that value.

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

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.

Precalculus is usually taken around 11th or 12th grade, 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 5: classify a discontinuity, where it is discontinuous, piecewise functions: continuous, jump, removable or infinite, the value that makes it continuous and the intermediate value theorem. 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.