discontinuity

A discontinuity is a place where the pen must leave the paper — a point of the domain at which the smooth flow of a function breaks. Something goes wrong: maybe the function jumps, maybe it has a hole patched with the wrong value, maybe it oscillates wildly or shoots off to infinity.

Precisely, f has a discontinuity at a point a of its domain when it fails to be continuous there: lim_{x->a} f(x) either does not exist, or exists but does not equal f(a). Discontinuities are traditionally sorted by how badly continuity fails. If both one-sided limits exist (and are finite), the discontinuity is of the first kind — either removable (the limits agree but miss f(a)) or a jump (they disagree). Otherwise it is of the second kind, where at least one one-sided limit fails to exist.

An honest caveat: ‘discontinuity’ is only meaningful at points actually in the domain. The function 1/x is not discontinuous at 0; 0 is not in its domain, so there is nothing to be discontinuous about. The set of points where a function jumps can be surprisingly large — a monotone function may have infinitely many jumps, though always only countably many.