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](http://mathworld.wolfram.com/MorerasTheorem.html), but I doubt their argument, specificly, the construction of the original function.



An extension of Morera's Theorem
The answer is yes, and a proof can be found for example on this webpage: <http://anhngq.wordpress.com/2009/07/20/a-generalization-of-the-morera%E2%80%99s-theorem/>


A brief summary: Suppose $f$ is continuous and $\int\_C f = 0$ for every circle $C$, but $\int\_\gamma f \neq 0$ for some closed curve $\gamma$. By convolving $f$ with a smooth approximation to the identity, we may assume $f$ is smooth. But then by applying Green's formula to $\int\_C f = 0$ for small circles $C$, we see that $f$ must satisfy the Cauchy-Riemann equations, so $\int\_\gamma f = 0$, a contradiction.


