AP Calculus AB and BC glossary

Continuity

Also called: Continuous function

A function is continuous at a point when three things hold: the function is defined there, the limit exists there, and the limit equals the function value. Informally, you can draw the graph through that point without lifting your pencil.

\lim_{x \to a} f(x) = f(a)

The compact statement $\lim_{x \to a} f(x) = f(a)$ quietly contains all three conditions: writing $f(a)$ requires it to exist, writing the limit requires it to exist, and the equals sign requires them to match. Exam questions often ask you to verify each part separately.

A function is continuous on an interval when it is continuous at every point of that interval, using one-sided limits at the endpoints. Continuity on a closed interval is the hypothesis that unlocks the Intermediate Value Theorem and the Extreme Value Theorem.

The mistake

Treating continuity and differentiability as the same thing. Differentiability implies continuity, but never the reverse: $f(x) = |x|$ is continuous at 0 and not differentiable there.

Appears in: Unit 1: Limits and Continuity