Approximate non-Lipschitz (but continuous) functions by Lipschitz functions

Question

Is there any algorithm to approximate non-Lipschitz (but continuous) functions by Lipschitz functions ?

Answer 1

Convolution with a mollifier, suggested by Jack D'Aurizio, works well on Euclidean spaces.

Another approach, which applies to general metric spaces, is to use the upper or lower $L$-Lipschitz envelope of given function $f:X\mapsto \mathbb R$. Specifically, $$g_L (x) = \sup_{y\in X}( f(y)-L\,d(x,y)) \tag{upper envelope}$$ $$h_L (x) = \inf_{y\in X}( f(y)+L\,d(x,y)) \tag{lower envelope}$$ For the above supremum and infimum to exist, we have to assume that $f$ has at most linear growth, that is $|f(x)|\le A+B\,d(x,x_0)$ for some constants $A,B$ and a point $x_0$. One-sided linear bound on $f$ suffices for one of these to exist.

Most of the following properties are straightforward to check:

1. Both $g_L$ and $h_L$ are $L$-Lipschitz functions
2. $h_L\le f\le g_L$ pointwise
3. $g_L$ is the infimum of all $L$-Lipschitz functions $g$ such that $g\ge f$ pointwise.
4. $h_L$ is the supremum of all $L$-Lipschitz functions $g$ such that $h\le f$ pointwise.
5. As $L\to\infty$, we have $g_L\to f$ and $h_L\to f$ pointwise

Perhaps the last one deserves a sketch of a proof. Fix $x\in X$ and $r>0$. When $L$ large enough, we have $f(y)-L\,d(x,y)<f(x)$ whenever $d(x,y)\ge r$. Therefore, the supremum in the definition of $g_L$ is effectively taken over the $r$-neighborhood of $x$. But in this neighborhood, when $r$ is small, the values of $f$ are all within $\epsilon$ of $f(x)$, and we conclude with $g_L(x)\le f(x)+\epsilon$.