A function $f$ on $\mathbb R^n$ is called harmonic if $\Delta f=0$ where $\Delta$ is the Laplacian. A function $f$ is harmonic if and only if $f$ satisfies mean value property (MVP), i.e. averaging of $f$ on any sphere is the value of $f$ at the centre. Can there be a non-harmonic function $f$ which satisfies MVP for spheres of some radii? In other words, to conclude that a function is harmonic do I have to test MVP for spheres of all positive radii (or at least on a dense set of radii)?