Do you know this function?

Solution 1:

Any $$f(x) = \frac{c + \sqrt{\vert x \vert}}{1+ a x^{2p}} \text{ or } \left(c + \sqrt{\vert x \vert}\right) e^{-a x^2}$$ with $a >0$ and $p \ge 1$ integer will do the job.

Approach:

• Find a function satisfying the first two criteria.
• Multiply it by a function that converges at $\infty$ fast enough to zero, that is positive and has a zero derivative at zero.
• Side element: notice that such a function will never be differentiable at zero.

Solution 2:

• the simplest solutions are $\frac{1}{x^\alpha+1}$ and $(1+x)^\alpha-x^\alpha$ with $\alpha \in (0,1)$
• every function of the form $f(x) = 1 - \sqrt{1-g(x)}$ with $g(0) = 1$ and $\lim \limits_{x \to \infty} g(x) = 0$ works, some examples are $g(x)=e^{-x}$ or $\frac{1}{x+1}$.
• some trig solutions are: $f(x) = \frac{\pi}{2} \operatorname{arccsc}(x+1)$ and $\exp(x^\alpha)$