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Math Practice

Continuity Practice Problems

A function is continuous at a when f(a) is defined, the limit exists, and the two are equal. Check those three conditions, and for piecewise functions make the pieces meet at the join.

  1. 1.

    Is continuous at ? Type yes or no.

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    Every polynomial is continuous everywhere.

  2. 2.

    Is continuous at ? Type yes or no.

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    is , undefined, so the first condition fails. (The limit exists, so it is a hole.)

  3. 3.

    At which value of is not continuous?

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    A rational function is continuous except where its denominator is zero: .

  4. 4.

    List every value of where is not continuous, separated by commas.

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    , which is zero at and .

  5. 5.

    Is for and for continuous at ? Type yes or no.

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    Left limit: . Right limit and : . All three conditions hold.

  6. 6.

    Is for and for continuous at ? Type yes or no.

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    Left limit: 3. Right limit: 2. The limit does not exist, so this is a jump.

  7. 7.

    A function has and . Is it continuous at ? Type yes or no.

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    The limit exists and is defined, but they are not equal, so the third condition fails.

  8. 8.

    On what interval is continuous? Use interval notation.

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    The root is defined for , and it is continuous on its whole domain, including the endpoint from the right.

  9. 9.

    Find so that for and for is continuous.

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    Set the pieces equal at : , so .

  10. 10.

    Find so that for and for is continuous.

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    At : , so and .

  11. 11.

    What value should be given so that becomes continuous at ?

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    The limit as is . Defining fills the hole.

  12. 12.

    Find using continuity.

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    is a composition of continuous functions, so substitute: .
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MPhil in Mathematics, Founder of Mathovia
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Continuity Practice Problems

These problems follow the continuity lesson.

The three-part test

A function is continuous at when all three hold:

  1. is defined.
  2. exists.
  3. .

Polynomials, rational functions, roots, trig, exponential and log functions are continuous wherever they are defined, so check only the special points.

Common mistakes to watch for

  • Checking only that the limit exists. The limit must also equal .
  • Checking one side at a join. Both sides must approach the value at the switch point.
  • Calling points outside the domain discontinuities. is continuous on its whole domain.
  • Ignoring endpoints. At a closed endpoint, use the one-sided limit from inside.

Frequently Asked Questions

How do I answer the yes or no questions?+

Type yes or no. Continuous and not continuous are also accepted.

How do I type an interval?+

Use interval notation such as [5, infinity) or (-infinity, 3]. A square bracket includes the endpoint.

More practice problems

One-Sided Limits Practice Problems Limit Laws Practice Problems Piecewise Functions Practice Problems