Morera's Theorem states that

If $f$ is continuous in a region $D$ and satisfies $\oint_{\gamma} f = 0$ for any closed curve $\gamma$ in $D$, then $f$ is analytic in $D$.

I have two questions:

1. If $f$ is continuous in $D$ and $\oint_C f = 0$ for any circle $C$ in $D$, can we deduce that $\oint_{\gamma} f = 0$ for any closed curve $\gamma$ in $D$?

2. (more ambitiously) If $f$ is continuous and $\oint_C f = 0$ for any circle $C$ in $D$, is $f$ analytic in $D$ ?

Partial ansers for question 2 seem to be here, but I doubt their argument, specificly, the construction of the original function.