Suppose that $f$, $g$ and $h$ are real valued continuous and strictly increasing functions defined on $[0,\infty)$ with value zero at zero. Furthermore, suppose that $f<g<h$ on all their domain, that $f$ and $h$ are locally Lipschitz at zero and that $g$ is continuously differentiable except in zero.
Can we claim that there exists a function $\hat{g}$ with the same properties as $g$ but locally Lipschitz at zero?