Cauchy-Goursat: If $f$ is analytic in a simply connected domain D, then $\int_{c}f(z) dz=0 $ for any closed curve $c$ in D.
However, I have seen questions where Cauchy-Goursat has been used when the function is just analytic in and on a closed curve C. Where are the simply connected regions in these situations? I know that Cauchy Gourmet can be applied to curves within C but why is it possible on C itself?