Hölder functions dense in space of bounded continuous functions (for non-compact manifolds)

Question

Let $M$ be a non-compact manifold and denote by $C_b(M)$ the space of bounded continuous functions on $M$. Is it true that the space of Hölder functions is dense in $C_b(M)$ (in the $C^0$ norm: $||f||=\sup_{x\in M}|f(x)|$)?

Answer 1

With $M = \mathbb{R}$, not even the uniformly continuous functions are dense in $C_b(\mathbb{R})$. As an example, take any bounded continuous functions with faster and faster oscillation, such as $x\mapsto \sin(x^2)$.

It's not hard to show that on any metric space---or more generally on any uniform space---the uniformly continuous functions are a $C^0$-closed subset of $C_b(M)$. So you can at most expect to be able to approximate these. I don't have access to the paper right now, but there is a version of the Stone-Weierstraß theorem due to Isbell that may give a useful sufficient criterion for when this actually happens.

Answer 2

It is not: for instance on the non-compact manifold $M:=\mathbb{R}_+$ the bounded continuous function $x\mapsto \sin(1/x)$ has obviously uniform distance greater or equal to $1$ from any uniformly continuous function, since the latter has a limit for $x\to0$.