= Continuous viscosity harmonic functions are classical
{title2=$\Delta u=0$}
Let a continuous function satisfy both viscosity inequalities for the <Laplace operator>. On any relatively compact ball, take its <harmonic replacement> $h$ with matching boundary values. If $u-h$ has a positive maximum, the function $h+\varepsilon(R^2-|x-a|^2)$ still lies below $u$ somewhere for sufficiently small positive $\varepsilon$, while matching it on the boundary. At an interior maximum of their difference it is a valid upper test, but its Laplacian is $-2n\varepsilon<0$, a contradiction. The opposite perturbation excludes a negative minimum. Thus $u=h$ throughout the ball and is a smooth <harmonic function>. A locally defined test is extended near its touching point using a smooth <cutoff function> when global tests are required.
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