Smooth approximation of Lipschitz functions on Riemannian manifolds

We show that for every Lipschitz function f defined on a separable Riemannian manifold M (possibly of infinite dimension), for every continuous , and for every positive number r>0, there exists a C∞ smooth Lipschitz function such that |f(p)−g(p)|⩽ε(p) for every p∈M and Lip(g)⩽Lip(f)+r. Consequently, every separable Riemannian manifold is uniformly bumpable. We also present some applications of this result, such as a general version for separable Riemannian manifolds of Deville–Godefroy–Zizler's smooth variational principle.