TITLE: Do there exist bump functions with uniformly bounded derivatives?
QUESTION [6 upvotes]: Let us consider a bump function $\phi: \mathbb{R} \longrightarrow \mathbb{R}$, smooth, with compact support. The most common examples are built from the function
$$ \psi(x) = \begin{cases} \exp ( \frac {1}{x^2 - 1}) & \lvert x \rvert < 1 \\
0 & \text{otherwise} \end{cases}.$$
Although this function behaves very well, its derivatives become arbitrarily large inside the unit ball. I wonder --- do there exist bump functions with "small" derivatives?

Stated more precisely, does there exist a bump function $\phi \in C^\infty_c(\mathbb{R})$ and some $M < \infty$ such that $\phi^{(k)} \leq M$ for all $k$th derivatives?

Intuitively, I feel the answer is no, as a sort of cost of vanishing entirely.

REPLY [5 votes]: If that bound held, then $\phi$ would be equal to its Taylor series everywhere (by considering the LaGrange form of remainder in Taylor's theorem), no matter where we center the series. In particular, we could center the series at a point where all derivatives are $0.$ That would give $\phi \equiv 0.$