[Math] Finding a bounded function with unbounded derivative at zero

Question

I am working on a linear analysis problem where we have boiled down the problem to finding a continuous function $f:\mathbb{R} \to \mathbb{R}$ that is bounded, but has infinite derivative at zero. So far, we have conjured up the example $$f_n(x) = \frac{2}{\pi}\arctan(nx)$$
This sequence of functions will have infinite derivative at $0$ when $n\to \infty$, and is bounded by $1$. I believe this will work for the sake of our problem, but I would like to find a function that doesn't depend on $n$. I can picture what this should look like, but I can't come up with an example function. Any ideas? All appreciated.

Answer 1

$f(x)=\arctan(\sqrt[3]{x})$, for example.