Smooth function with all derivatives zero
Assume that $f\colon\mathbb{R\to R}$ is an infinitely often differentiable function, and that a point $a\in\mathbb R$ satisfies $f^{(k)}(a) = 0$ for alle $k\ge 1$. Does there exist an $\epsilon > 0$ such that $f$ is constant at $(a-\epsilon, a+\epsilon)$? If not, can somebody find a counterexample?
Solution 1:
There is a standard example: $f(x) = e^{-1/x}$ when $x >0$ and $f(x) =0$ when $x\leq 0$.