Thread: Limit question

Suppose f is Complex differentiable everywhere and that Lim|z|→∞}f(z)=1
Prove f(z) is constant

You have to use Liouville theorem. Find a closed ball B such that outside the modulus of f is smaller than 2, and apply that the closed ball is compact to conclude that f is bounded-

Originally Posted by thespian

Suppose f is Complex differentiable everywhere and that Lim|z|→∞}f(z)=1
Prove f(z) is constant

First proof: Liouville's theorem states that a holomorphic (analytic) function which is bounded must be constant, and that's exactly what happens with our function: in some circle

Second proof: By Cauchy's integral representation and by the estimation lemma: , so constant.

Tonio