Let $p>1$ and let $g:\mathbb{R} \to \mathbb{R}$ given by $g(t)=|t|^{p-1}$ if $t \geq 0$ and $g(t)=-|t|^{p-1}$ if $t<0$. It is possible to construct an sequence $(g_m)_{m=1}^{\infty}$ of continuously differentiable odd increasing functions such that $g_m \to g$ uniformly on $\mathbb{R}$?
The reason I'm searching such approximation is that if this is possible I will be able to understand a nice regularity result for an Elliptic PDE, any help will be very useful, thank you.