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Limits & Continuity · Continuity & Theorems

Continuity Definition

f is continuous at a  ⟺  lim⁡x→af(x)=f(a)f \text{ is continuous at } a \iff \lim_{x \to a} f(x) = f(a)

A function is continuous at a point a if (1) f(a) is defined, (2) lim(x→a) f(x) exists, and (3) the limit equals f(a).

  • 2 worked examples
  • 18 in Limits & Continuity

Worked examples

Is f(x) = (x²-4)/(x-2) continuous at x = 2?
  1. f(2) is undefined (0/0), so condition 1 fails

Answer: No, f is not continuous at x = 2 (not defined there).

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Is f(x) = x² + 1 continuous at x = 3?

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Related formulas

Intermediate Value TheoremExtreme Value Theorem

See all 18 Limits & Continuity formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Common questions

What is the Continuity Definition?

A function is continuous at a point a if (1) f(a) is defined, (2) lim(x→a) f(x) exists, and (3) the limit equals f(a). It is one of the limits & continuity formulas in the CalcRef reference.

How do you use the Continuity Definition?

Worked example. Is f(x) = (x²-4)/(x-2) continuous at x = 2? f(2) is undefined (0/0), so condition 1 fails. Answer: No, f is not continuous at x = 2 (not defined there).

Practice

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