Let $C^\infty$ denote the collection of functions $f:\mathbb{R}\to\mathbb{R}$ such that for every positive integer $n$, the $n$-th derivative of $f$ exists. For $f\in C^\infty$ we set

- $f^{(0)} = f$, and
- $f^{(n+1)} = \big(f^{(n)}\big)'$ for all non-negative integers $n$.

Is there $f\in C^\infty$ with the following properties?

- for all $x\in (-\infty, 0]$ we have $f(x)=0$.
- for all non-negative integers $n$ and $x\in (0,\infty)$ we have $f^{(n+1)}(x) > f^{(n)}(x)$
- there is a function $h:\mathbb{R}\to\mathbb{R}$ such that for all non-negative integers $n$ and all $x\in\mathbb{R}$ we have $f^{(n)}(x)\leq h(x)$.

(Additional question for curiosity, answering it is not needed for acceptance of answer: can $h$ be chosen to be continous? Or even $h\in C^\infty$?)