Let $G \subset \mathbf{C}$ be an open set. A continuous function $u:G \to \mathbf{R}$ is said to have the MVP (Mean Value peoperty) if whenever $\overline{B(a,r)} \subset G$ we have $$u(a)=\frac{1}{2\pi}\int_0^{2\pi} u(a+re^{it})dt \,\,\,\,\,\, \dagger$$
A continuous function $u:G \to \mathbf{R}$ is said to have the local MVP if for each $a \in G$ there exists $R>0$ such that for all $0 \leq r \leq R$ we have $$u(a)=\frac{1}{2\pi}\int_0^{2\pi} u(a+re^{it})dt \,\,\,\,\,\, \ddagger$$
Conway proves any continuous function with MVP must be harmonic by means of solving Dirichlet problem in a ball and maximum principles. Clearly, MVP $\implies$ Local MVP. But how can I show Local MVP $\implies$ MVP? I tried tweaking around with maximum principles and Poisson integral formula for solving Dirichlet Problem in a ball but nothing worked.
Any hints would be greatly appreciated.
Note: This has been asked multiple times here but I could not find a complete solution anywhere and in my setting where the domain is $\mathbf{C}$