If $f:\Bbb C \to \Bbb C$ is continuous such that $f$ is analytic outside the bounded set $S$ then prove that $f$ is entire.
To prove this we have to prove that $f$ is analytic in $S$. For this I I think it can be done by using Morera's theorem , which states that " If $f$ is continuous in a simply connected domain $D$ and $\int_Cf(z)\,dz=0$ , for every closed contour $C$ in $D$ then $f$ is analytic in $D$."
But here how I can show that $\int_C f(z)\,dz=0$ for every closed contour $C$ in the bounded domain $S$?
I've seen this similar type question , but there are no enough proof which I want. So , please don't make this question as a duplicate question , and give me some hints to show the integral value is $0$.