Suppose I have an open set $U$ in the complex plane and a function $g$ that is continuous on $U$.
Let $C(z_0$$,r)$ be a circle fully contained in $U$ of radius $r$ whose center is $z_0$.
I know that if $g$ is harmonic, then the mean value of $g$ over $C(z_0$$,r)$ is equal to $g(z_0)$.
But suppose we do not require $g$ to be harmonic.
Does $g(z_0)$ equal the limit as $r$ tends to $0$ of the mean values of $g$ over the circles $C(z_0$$,r)$?
I'd appreciate any suggestions on how to prove this result or some reading material where I might find the answer.
You should be able to prove this result using pretty much just the definition of continuity.
If $g$ is continuous and we take $\epsilon > 0$ then for all $z$ sufficiently close to $z_0$, we have $| g(z) - g(z_0) |< \epsilon$, from which you should be able to show that the mean on any sufficiently small circle (centred at $z_0$) is within $\epsilon$ of $g(z_0)$.