Find all functions $f$ that satisfy the following

Question

Let $\Omega $ be an open, bounded, and connected subset of $\mathbb{C}$

Find all functions $f:\bar{\Omega }\rightarrow \mathbb{C}$ that satisfy the following conditions simultaneously :

1. $f$ is continuous

2. $f$ is holomorphic on $\Omega $

3. $f(z)=e^z$ for all $z\in \partial\Omega$

My work: $e^z$ is analytic and $f(z)=e^z$ on $\partial\Omega$ which is closed , then it contains all of its accumulation points.

Then for all $z\in \partial\Omega $ , $f(z)=e^z$ on a neighborhood of $z$ and $Germ(f-e^z,z)=0$

Then by principle of analytic continuation $f(z)=e^z$ on $\bar{\Omega }$

Correct ?

Answer 1

Your argument is not valid because $f$ is not holomorphic in a neighborhood of $z\in \partial\Omega$.

But you can apply the maximum modulus principle to the difference $g(z) = f(z) - e^z$ and conclude that $g$ must be identically zero in $\Omega$.