In Chapter 11 of Rudin's RCA, a harmonic function is defined to be a complex continuous function $u$ on a plane open set such that the Laplacian of $u$, i.e. the sum of its pure second-order partial derivatives $$u_{xx}+u_{yy}$$ is $0$. I was wondering if this was missing a condition, namely that $u_{xy}=u_{yx}$, because I was unable to show the following without this assumption.
For every harmonic function $u$ whose domain includes the image of a holomorphic function $f$ in a plane open set $\Omega$, the composite $u\circ f$ is harmonic in $\Omega$.
I attempted to show this by just brute calculation, and what remained was $u_{xy}-u_{yx}$ with some partials of $f$ multiplied on the outside.
Of course, Rudin later shows that harmonic functions have continuous partial derivatives of all orders because the real-valued ones are locally real parts of holomorphic functions, but I think his proof relies on a certain composite of functions being harmonic...
You don't need to assume equality of the mixed partials in the definition. Here is a rough outline of one proof:
First, show the maximum principle for harmonic functions. Note that this implies that two functions which are continuous on a closed disk and harmonic in the disk's interior, and equal on the boundary, are equal in the interior as well.
Second, note that the Dirichlet problem on a disk with continuous boundary value has an explicit solution given by the separation of variables method in polar coordinates, and this solution is $C^\infty$.
Combine the above to show that any harmonic function is $C^\infty$ everywhere. In particular, mixed partials, being continuous, are equal.