Continuity (calculus)

A function is continuous when its graph can be drawn without lifting the pencil — no holes, jumps, or vertical asymptotes. Formally, a function f is continuous at a point x = c when three conditions hold: f(c) is defined, the limit of f(x) as x approaches c exists (the left-hand and right-hand limits agree), and that limit equals f(c). If any condition fails, the function is discontinuous at that point.

Discontinuities come in recognizable types. A removable discontinuity is a single hole, as in f(x) = (x² − 4)/(x − 2), which looks like the line y = x + 2 with a hole at x = 2. A jump discontinuity occurs when the one-sided limits exist but disagree, as in many piecewise functions. An infinite discontinuity occurs at a vertical asymptote, where the function grows without bound.

Continuity matters because the major theorems of calculus require it. The Intermediate Value Theorem — if f is continuous on [a, b], it takes every value between f(a) and f(b) — and the Extreme Value Theorem both depend on continuity over a closed interval, and differentiability at a point requires continuity there (though a continuous function, like |x| at 0, need not be differentiable).

The AP Calculus AB exam tests continuity directly: classifying discontinuities, choosing a constant that makes a piecewise function continuous, and applying the three-part definition. It also tests continuity as the hypothesis behind the Intermediate Value Theorem and Extreme Value Theorem, so know both the definition and the theorems that rely on it.