Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be non-constant and differentiable with a sequence of $(x_n)_{n \in \mathbb{N}}$ such that $f'(x_n)=0 \, \forall \, n$ and $x_n \rightarrow c$ for some $c \in \mathbb{R}$. Can $f$ be analytic?
I'm looking for either a proof that $f$ cannot be analytic or an example of an analytic $f$.
(I'm most of the way through an answer to this question, trying to prove that an analytic function of a continuous random variable is itself a continuous random variable. But I'm stuck and need to prove that an $f$ as described cannot be analytic.)
No, such a function cannot be analytic, unless it is constant. If it was analytic, $f'$ would be analytic too. But $f'(c)=\lim_{n\to\infty}f'(x_n)=0$. Therefore, by the identity theorem, $f'$ is the null function and so $f$ is constant.