I realize that what I did in the previous post was, well, lame.
The setting: let be continuous but non-analytic in some disk in the complex plane, and let be analytic in which, for the purposes of this informal note, we will take to contain an open disk. If doesn’t contain an open set or if the partials of fail to exist, the question of being analytic is easy and uninteresting.
Let and where are real valued functions of two variables which have continuous partial derivatives. Assume that and (the standard Cauchy-Riemann equations) in the domain of interest and that either or in our domain of interest.
Now if the composition is analytic, then the Cauchy-Riemann equations must hold; that is:
Now use the chain rule and do some calculation:
From the first of these equations:
By using the C-R equations for we can substitute:
This leads to the following system of equations:
This leads to the matrix equation:
The coefficient matrix has determinant which is zero when BOTH and are zero, which means that the Cauchy-Riemann equations for hold. Since that is not the case, the system of equations has only the trivial solution which means which implies (by C-R for ) that which implies that is constant.
This result includes the “baby result” in the previous post.