Approximation of a continuous function by a smooth one on an open set

Question

I'm interested in the following kind of theorems :

Let $M$ be a real analytic manifold and $U$ an open set of $M$. Let $f : U \to \mathbb{R}$ a continuous function. Then, there is a $C_{\infty}$ function $g : U \to \mathbb{R}$ and a constant $C>0$ such that $$|f(x)-g(x)| \leq C, \quad \forall x \in U.$$ I would like to know what can be standard (or less common) constructions of such $g$ and $C$. Note that I don't put any condition on the open set but I also don't ask a true approximation (i.e. for all $\epsilon$ as small as one wants). I just want the inequality to be true for a specific constant.

If one needs more specific conditions on $U$ and $f$, it can also be interesting for me but I would like to have the more general constructions that it is possible to have. Thank you in advance for any help or references.

Answer 1

Whitney's various approximation theorems, using convolution in local coordinates locally, and then patching, acheive results like the one you want. See Hirch, Differential Topology, p. 45, theorem 2.2: on any smooth manifold, smooth functions are dense (in the fine topology) in continuous functions. You simply throw away the manifold $M$ and replace it by the open set $U$.