Complex Calculus question relating to Moreras theorem.

We have a ton of theorems for complex functions that are analytic on a domain D. We don't have very many iff statements in fact the one that i do have is stated as a totally different theorem.

Morera's theorem: let f be continuous on a domain D,

if $ \int_{C} f(z) dz =0 $ for every closed contour C in D then f is analytic throughout D.

I have found this theorem really useful for any well finite domain D is there some way we can apply this to say that for example $f(z)= z^c$ where $ z^c $ is the complex conjugate (cant seem to figure out the bar language.)

i know that $f(z)=z^c$ is continuous on the complex plane if there some way to apply this theorem to show that it is not analytic on C?

I guess what im asking is like the converse of the principle of deformation of paths for f continuous true?

Answer

Let $C=\{z \in \mathbb C:|z|=1\}$. Then, since $z \overline{z}=1$ for $z \in C$:

$\int_{C} \overline{z} dz=\int_{C} \frac{1}{z} dz=2 \pi i \ne 0$