Let $C$ be the subset of $C_b(\mathbb{R})$ given by $$C:=\{f\in C_b(\mathbb{R}):\ \exists f'\in C_b(\mathbb{R})\}$$ Now I want to take the closure of this set with respect to the supremum norm on $C_b(\mathbb{R})$. Hence the closure contains functions which can be uniformly approximate by sequences of functions (f_n)_{n\in\mathbb{N}} such that $f_n\in C_b(\mathbb{R})$ and $f'_n$ exists for every $n\in\mathbb{N}$ in $C_b(\mathbb{R})$. Hence the limit function has first to be bounded since it is the uniform limit of bounded functions. Since every derivate is bounded, we know that $f_n$ is uniformly continuous. But what can I say more about the elements in this closure? Is there a explicit description of this closre by well-know spaces, for example $W^{2,2}(\mathbb{R})$ (which will not be true here)?

Thank you very much.

Let $C$ be the subset of $C_b(\mathbb{R})$ given by $$C:=\{f\in C_b(\mathbb{R}):\ \exists f'\in C_b(\mathbb{R})\}$$ Now I want to take the closure of this set with respect to the supremum norm on $C_b(\mathbb{R})$. Hence the closure contains functions which can be uniformly approximate by sequences of functions $(f_n)_{n\in\mathbb{N}}$ such that $f_n\in C_b(\mathbb{R})$ and $f'_n$ exists for every $n\in\mathbb{N}$ in $C_b(\mathbb{R})$. Hence the limit function has first to be bounded since it is the uniform limit of bounded functions. Since every derivate is bounded, we know that $f_n$ is uniformly continuous. But what can I say more about the elements in this closure? Is there a explicit description of this closre by well-know spaces, for example $W^{2,2}(\mathbb{R})$ (which will not be true here)?

Thank you very much.