Are all complex differentiable functions continuous?

Question

If a function f, which maps an open subset A of the complex plane into the complex plane, is complex differentiable on A, does that mean it's continuous on A? I know that this holds for real-valued functions, but was curious if there was something about complex-valued functions that made things different.

Answer 1

Yes, differentiable implies continuous.

Complex-valued does make a difference: if your function has a continuous derivative, it would be holomorphic (analytic), which is of course not true for real variables.

Steve

Answer 2

In complex we discuss whether the function is analytic not continuos