Let f : open unit ball -> C be continuous.

Suppose that f is (complex)differentiable for all z in the unit ball with f(z) not equal to 0.

Prove that f is holomorphic.

The question reduces to simply the case if for all balls around z0 with f(z0)=0, we have some zero and non-zero f(z). The other cases are obvious

If f is holomorphic this isn't possible since the zeros are isolated, but I'm stumped on how to show this.

With Cauchy-Goursat it's possible if there was only one f(z)=0.

Any suggestions?