Assume $u$ is continuous such that $u(x)=\frac{1}{|B(x,r_x)|}\int_{B(x,r_x)} u(y)dy$. Is $u$ harmonic?

Question

Assume $u:\Omega\subset\mathbb{R}^d\to\mathbb{R}$ is continuous and satisfies the property: for every $x\in \mathbb{\Omega}$ there is $r_x>0$ such that $$ u(x)=\frac{1}{|B(x,r_x)|}\int_{B(x,r_x)} u(y)dy,\hskip .1in\text{(Discrete Mean-value property).}\tag{1}$$ what can we say about the harmonicity of u? which kind of harmonicity do we have? $$ $$ Recall that if (1) holds independently on $r_x$ (that is for every $r>0$ such that $B(x,r)\subset \Omega$), then $u$ is harmonic.

Answer 1

Hint for a partial result: Suppose $u$ is continuous on the closed unit ball $\overline B.$ If for every $x\in B$ there exists $B(x,r_x),$ with $r_x \le 1-|x|$ such that your the value condition holds, then $u$ is harmonic in $B.$ Proof idea: Consider $u-v, $ where $v$ is the Poisson integral of $u|_{\partial B}.$