Let $f:\Omega\subset\Bbb{R}^m\to\Bbb{R}^n$ be a given function. In general, how can I show that $f$ is smooth (infinitely differentiable) and analytic. I know this is a bit vaguely stated question, but I just need a general idea about showing smoothness and analyticity.
If you can give me some examples that would be great. PDF's and other sources are also welcome.
A possible approach is to
guess what the power series development of $f$ is,
show that the function $g$ defined via this development is indeed analytic (converges on a non-trivial domain),
show that $g$ shares enough properties with $f$ that they are the same function (e.g. both are solutions to a differential equation with boundary conditions).