For every closed ball contained in the domain of a harmonic function, the value at its centre equals both its average over the ball and its average over the boundary sphere.
If a continuous function has the spherical mean value property at every point for some sequence of radii decreasing to zero, then it is harmonic. Comparison on each relatively compact ball with the harmonic function having the same boundary data proves the result by propagating any hypothetical interior extremum along one of the admissible spheres.
If a harmonic function on a connected domain attains an interior maximum or minimum, it is constant. The mean value property forces equality throughout every sufficiently small ball around an interior extremum, and connectedness propagates the equality.
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