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I know that u is a continuous function where

$$u(x_0) $$ is equal to the average over the surface of all balls centered at a point $x_0$.

How would I show that this function is harmonic, or approach this problem. Would I find a harmonic function $f$ which is equal to $u$ on the boundary and just do something with the strong maximum principle?

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Your approach is relevant. Indeed, you restrict yourself in a closed ball contained in the open set and construct a function $f$ which is harmonic in the open ball and which agrees with $u$ in the boundary of the ball. This is done by the Poisson integral. Once this is done, you use the mean property of $u$ to prove that $u = f$. Since Harmonicity is obviously a local property, this gives you the conclusion.

This is done in Walter Rudin, Real and Complex analysis. Check Theorem 11.16, page 230.

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