Continuity

Learn continuity at an input, jump discontinuities.

Lesson 3 of 32 parts Practice Flashcards

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Continuity at an Input

Objective: Explain continuity at an input, apply the stated method, and check its conditions.

A function is continuous at a when f(a) is defined, its finite two-sided limit exists, and the limit equals f(a). Assigning the limiting value can repair a removable discontinuity. A jump cannot be repaired by changing one value alone. All three conditions—value, limit, and agreement—are required.

Formulas

$limₓ→a f(x)=f(a)$

Worked examples

Try itLet f(x)=(x²−49)/(x−7) for x≠7 and f(7)=k. Find k for continuity.Show answer

Step 1: For nearby x, simplify to x+7. Step 2: The two-sided limit is 14. Step 3: Continuity requires k=14.

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Continuity at an Input

Objective: Explain continuity at an input, apply the stated method, and check its conditions.

Formulas

$limₓ→a f(x)=f(a)$

Worked examples

Try itLet f(x)=(x²−49)/(x−7) for x≠7 and f(7)=k. Find k for continuity.Show answer

Step 1: For nearby x, simplify to x+7. Step 2: The two-sided limit is 14. Step 3: Continuity requires k=14.

Jump Discontinuities

Objective: Explain jump discontinuities, apply the stated method, and check its conditions.

A jump occurs when finite one-sided limits differ. The function can have a defined value at the target without a two-sided limit. Choose each side’s formula from the neighboring inputs, not merely the rule defining the value at the endpoint. The assigned point value cannot force unequal one-sided limits to agree.

Formulas

If L₋≠L₊, the two-sided finite limit does not exist

Worked examples

Try itf(x)=6x+7 for x<7 and f(x)=6x+8 for x≥7. Find the right-hand limit at 7.Show answer

Step 1: The right-hand side uses 6x+8. Step 2: Approach x=7 in that expression. Step 3: The right-hand limit is 50; it differs from the left limit.

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