TITLE: Existence of an infinitely differentiable function $ f $ with $ {f^{(n)}}(0) = 0 $ for all $ n \in \mathbb{N} $.
QUESTION [5 upvotes]: How can one show that there exists an infinitely differentiable function $ f: \mathbb{R} \to \mathbb{R} $ such that $ {f^{(n)}}(0) = 0 $ but $ f^{(n)} \not\equiv 0 $ for all $ n \in \mathbb{N} $?

REPLY [6 votes]: The function $f$ given by
$$
f(x)=
\begin{cases}
\exp\left(-1/x\right),\quad&x>0,\\
0,&x\leq 0
\end{cases}
$$
has derivatives of all orders that satisfy $f^{(n)}(x)=0$ for $x\leq 0$ and $n\in\mathbb N$. See e.g. wikipedia and/or this question.