Continuous function that approaches 0 slower than any $x^{\alpha}$

Question

Is there a continuous function $f$ with $f(0) = 0$ such that $$\sup_{\alpha \in \mathbb{R}} \left[ \lim_{x\to 0^{+}} x^{-\alpha} f(x) < \infty \right] = 0?$$

(I'm guessing there probably is? What's an example of such a function?)

Answer 1

Let $f(x)=e^{-1/x^{2}}$, $x\ne 0$, and $f(0)=0$, then $\lim_{x\rightarrow 0^{+}}x^{-\alpha}f(x)=0$ for any $\alpha\in{\bf{R}}$.