Continuity

Math

Definition

A function is continuous at a point x = a if three conditions are met: f(a) is defined, the limit as x approaches a exists, and the limit equals f(a). Informally, you can draw the function at that point without lifting your pencil.

Examples

  • f(x) = x² is continuous everywhere
  • f(x) = 1/x is not continuous at x = 0 because f(0) is undefined
  • A piecewise function may be discontinuous where the pieces meet if the values don't match
Key Fact

Continuous at a: (1) f(a) exists, (2) lim x→a f(x) exists, (3) lim x→a f(x) = f(a)

Related Terms

Limit definition Ivt Squeeze theorem Piecewise functions

Study This Concept

Practice continuity with free review games in these units:

AP Calculus AB Unit 1: Limits and Continuity