Suppose $f,g: \mathbb{R} \to \mathbb{R}$ are real analytic, i.e, locally given by convergent power series. Then $g \circ f$ is real-analytic as well.
How do I prove this? I guess the "standard" proof would be to extend $f$ and $g$ into some open subsets of $\mathbb{C}$ in the natural way via their power series, then notice that $g \circ f$ is complex differentiable, hence complex-analytic, and hence real-analytic when restricted to $\mathbb{R}$.
But is there another proof of this, one that doesn't use complex-analytic extensions?
I want a proof which can be extended to the multivariate case, i.e, if $f: \mathbb{R}^n \to \mathbb{R}^m$ and $g:\mathbb{R}^m \to \mathbb{R}^p$ are both analytic (i.e, their components are locally given by multivariate power series), then $g \circ f$ is also real-analytic. Is the standard proof for the multivariate case via complex-analytic extensions as well? Does anyone know a good book for this subject?
Best Answer
A good reference for this stuff is A Primer of Real Analytic Functions by Steven Krantz and Harold Parks.