continuity

Continuity is the mathematical version of drawing a curve without lifting your pen: nearby inputs go to nearby outputs, with no sudden jumps, gaps, or holes. The graph flows; you can trace it through the point without a break.

Formally, f is continuous at a (a point of its domain) if lim_{x->a} f(x) = f(a). Unpacking the limit, this means: for every epsilon > 0 there is a delta > 0 such that |x - a| < delta and x in the domain imply |f(x) - f(a)| < epsilon. Compared with the bare limit definition, the strict 0 < |x - a| is dropped, and the limit value is required to equal f(a) — so continuity demands three things at once: f(a) is defined, the limit exists, and they coincide.

Continuity at a is automatic and meaningless to test at an isolated point of the domain (every function is continuous there), and at a cluster point it reduces exactly to the limit condition. The familiar functions of calculus — polynomials, exponentials, sines, and their sums, products, quotients, and compositions on their domains — are all continuous, which is why we rarely notice the condition until it fails.