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Calculus practice

Determining Continuity

Practice exact continuity intervals and one-sided-limit reasoning for piecewise, absolute-value, kinked-polynomial, and rational functions.

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Find maximal open intervals where the function is two-sided continuous. Use exact one-sided limits and function values at every critical point.

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Test each critical point exactly

Students compare left and right limits with the defined value, then report maximal open continuity intervals.

Learning objective

What this activity builds

Find maximal open intervals of two-sided continuity and distinguish holes, jumps, poles, and continuous corners.

Worked example

Does a corner split a continuity interval?

No. A corner can be nondifferentiable while the function remains continuous.

Continuity depends on matching limits and function values, not on matching derivatives.

Teaching notes

Make the practice count

  • Keep original denominator restrictions after cancelling factors.
  • Compare both one-sided limits with the defined point value.
  • Check residual pole multiplicity before assigning infinity signs.

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Math SheetNameScoreTeacherDateDetermining ContinuityFind maximal open intervals where the function is two-sided continuous. List each discontinuity and classify it as removable, jump, or infinite.1. Find intervals; identify and classify discontinuities.f(x) =-x-1+2·x-52Intervals:Discontinuities / types:2. Find intervals; identify and classify discontinuities.f(x) =2·x+12x+12Intervals:Discontinuities / types:3. Find intervals; identify and classify discontinuities.f(x) =2x < -3/2x+112x > -3/2f(-3/2) = 2Intervals:Discontinuities / types:4. Find intervals; identify and classify discontinuities.f(x) =x+4Intervals:Discontinuities / types:Intervals use two-sided continuity on maximal open spans.math-sheet.comPage 1 of 2
Math SheetNameScoreTeacherDateDetermining ContinuityFind maximal open intervals where the function is two-sided continuous. List each discontinuity and classify it as removable, jump, or infinite.5. Find intervals; identify and classify discontinuities.f(x) =-2·x2+3-x+32Intervals:Discontinuities / types:6. Find intervals; identify and classify discontinuities.f(x) =-12·x-32x-12Intervals:Discontinuities / types:7. Find intervals; identify and classify discontinuities.f(x) =-1·xx < -3/2-2·x-32-3/2 < x < -1/2-x-1x > -1/2f(-3/2) = undefinedf(-1/2) = 1/2Intervals:Discontinuities / types:8. Find intervals; identify and classify discontinuities.f(x) =-2·x+4Intervals:Discontinuities / types:Intervals use two-sided continuity on maximal open spans.math-sheet.comPage 2 of 2

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