Suppose $y:[0, \infty)\to [0,\infty)$ is differentiable with $y(0)=0$ and such that for every $\delta>0$ we have that $y|\_{0,\delta}\not \equiv 0$.

Does it follow that there exists $\varepsilon>0$ is strictly monotonic over $[0,\varepsilon)$?

This seems intuitively true, but I was not able to prove it.