Charecterize all complex valued functions $f$ satisfying below statement:
Suppose $C$ is a simple closed curve inside a simply connected domain $D$ and for any point $a$ inside $C$, $$2i\pi f(a)=\int_C \frac{f(z)}{z-a}\,dz$$
Characterizing means finding whether $f$ is analytic inside $C$, or whether it has any singularities inside $C$ , or it is not analytic inside $C $ etc.
I try to prove that $f$ is analytic inside $C$ mimicking the proof of Cauchy integral formula for $f'(a)$ but I get stuck in some steps as we do not even know whether $f$ is continuous. Any help is appreciated.