sequential criterion for continuity

There are two ways to test whether a function never jumps. One uses epsilon and delta; the other uses sequences. The sequential criterion says: a function is continuous at a point exactly when it carries every sequence converging to that point onto a sequence converging to the value there. Continuity is precisely the preservation of convergence.

Stated precisely, f is continuous at a if and only if for every sequence (x_n) in the domain with x_n -> a, we have f(x_n) -> f(a). Note that here, unlike for the functional limit, the x_n are allowed to equal a; continuity does not exclude the point itself.

This criterion is a workhorse for proving discontinuity: to show f is not continuous at a, exhibit a single sequence x_n -> a for which f(x_n) fails to converge to f(a). It also cleanly transports limit laws from sequences to functions, and it is the bridge to the more general ‘sequential continuity’ studied in metric spaces, where it coincides with ordinary continuity.