isolated point

An isolated point of a set is a lone outpost — a member of the set surrounded by empty space, with no other points of the set anywhere nearby. You can draw a small circle around it that catches nothing else from the set.

Precisely, a point p of a set S is isolated if there is some radius r > 0 such that the only point of S within distance r of p is p itself: the neighborhood (p - r, p + r) meets S in {p} alone. Equivalently, p belongs to S but is not a cluster point of S — there is no sequence of other points of S approaching it.

Isolated points matter for continuity in a quietly important way: a function is automatically continuous at every isolated point of its domain, with nothing to prove. There is no nearby behavior to compare against, so the epsilon-delta condition is vacuously satisfied (any small enough delta isolates the point). This is why the substantive content of continuity lives entirely at cluster points.