Let $f(x,y)$ be a differentiable and integrable function defined in a domain $D\subset\mathbb{R}^2$.
$\forall(x_0,y_0)\in D$, $C$ is a circle in $D$ with center $(x_0,y_0)$ and arbitrary radius $r$. We are given that
$\displaystyle f(x_0,y_0) = \frac{1}{2\pi}\oint_C f(x,y)d\theta$
Equivalently, the function value at a point is average of values at circle with center at that point, and arbitrary radius such that circle is fully inside the domain $D$.
What notable observations can we make about $f(x,y)$? I know that if $f(x,y)$ is harmonic, it satisfies the mean-value property, but I'm interested in the converse.
Also curious about higher dimension generalization.