True or False? If $f(z)$ is analytic in a simply connected domain $D$ and continuous in $\bar{D}$, then $\oint_{\delta D}f(z)dz=0$.

Question

If $f(z)$ is analytic in a simply connected domain D and continuous in $\bar{D}$, then $\oint_{\partial D}f(z)dz=0$.

I think we can say this is true but I am having a hard time forming a proof other than since it is continuous on the boundary, we can't have any new singularities from the analytic domain to the boundary.

Answer 1

The answer to the question is "yes" subject to some geometric details. There is an explanation at https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem. Here's the relevant paragraph.

Unfortunately, I don't have access to the Kodaira reference.