Find all functions $f$ that satisfy the following

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 $zin partialOmega$

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

Then for all $zin partialOmega $ , $f(z)=e^z$ 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 ?