I have an increasing continuous function $f:{\mathbb R}\rightarrow {\mathbb R}$ which is not differentiable everywhere, and I would like to approximate it with an infinitely differentiable function $g\in C^{\infty}$. I found this paper:

Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds. Arxiv,

which tells me I can do it uniformly. However, nothing tells me that $g$ is also increasing, and this property is essential in my context. I would appreciate any pointers on the existence of a uniform approximation to $f$ in $C^{\infty}$ which is also increasing.

I have an increasing continuous function $f:{\mathbb R}\rightarrow {\mathbb R}$ which is not differentiable everywhere, and I would like to approximate it with an infinitely differentiable function $g\in C^{\infty}$. I found this paper:

Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds. Arxiv,

which tells me I can do it uniformly. However, nothing tells me that $g$ is also increasing, and this property is essential in my context. I would appreciate any pointers on the existence of a uniform approximation to $f$ in $C^{\infty}$ which is also increasing.

P.S. My function is basically: $K_1 x I(x<0) + K_2x I(x\geq 0)$, $K_1,K_2>0$.