Increasing Function bounded below by a polynomious Suppose $g:[0,\infty)\rightarrow [0,\infty)$ is a cotinuous increasing function with $g(0)=0$. Is it possible to find two constants $C&gt;0$ and $q&gt;0$ such that $$g(s)\geq Cs^q,\ \forall\ s\in [0,\delta)$$

where $\delta&gt;0$ is an small number.  

Thank you.

Edit: Sorry, I forgot one hypothesis.
This is still not true. Recall the classical example of a smooth non-zero function with all Taylor coefficients vanishing at 0 (you can construct it with $e^{-1/x}$ for $x&gt;0$ and 0 otherwise).

The Taylor Young theorem says that you won't be able to find such constants for that function.
No,. this need not be true. One simple way it could fail is if $g$ is bounded (since $Cs^q$ is unbounded). So for example $g(s)=\arctan(s)$ is a counterexample.