Liouville's theorem (complex analysis)

The theorem is considerably improved by Picard's little theorem, which says that every entire function whose image omits two or more complex numbers must be constant.

A standard analytical proof uses the fact that holomorphic functions are analytic.

Given two points, choose two balls with the given points as centers and of equal radius. If the radius is large enough, the two balls will coincide except for an arbitrarily small proportion of their volume. Since f is bounded, the averages of it over the two balls are arbitrarily close, and so f assumes the same value at any two points.

The proof can be adapted to the case where the harmonic function f is merely bounded above or below. See Harmonic function#Liouville's theorem.