How to show real analyticity without extending to complex plane Suppose we have some $f \in C^\infty(\mathbb{R},\mathbb{R}).$  For example, $$f(x)=(1+x^2)^{-1}.$$  Using complex analysis, we can easily show $f$ is real analytic.  Is there an easy, general method to show this which doesn't use complex analysis?

Sums, products, quotients, and compositions of real analytic functions are real analytic. So in your example, $1+x^2$ is real-analytic, and so is $1/x,$ hence so is $1/(1+x^2).$
