$X$ second countable locally compact Hausdorff implies $C(X)$ separable? In the post When is $C_0(X)$ separable? , it is argued that if $X$ is second countable and locally compact Hausdorff, then $C_{0}(X)$ is separable. Is it also true that $C(X)$ is separable under the same hypotheses?
 A: It is not true in general. For example $C(\mathbb{R})$ is not separable because you can find uncountable familily of continuous functions with pairwise distance equal to $1$. Below I present its simple counstrucion. Take arbitrary binary sequence $w\in\{0,1\}^\mathbb{N}$ and consider continuous function
$$
f_w(x)=\sum\limits_{k=1}^\infty w_k\max(1-|2x-k|,0)
$$
It consists of triangle shaped peaks at point $k/2$ if $w_k=1$ and plato plateau if $w_k=0$. The desired family of functions is $\{f_w:w\in\{0,1\}^\mathbb{N}\}$. 
A: It seems that even $C(\mathbb R)$ or ($C(\mathbb Z)$) in the topology 
of uniform convergence is not separable.  Suppose the opposite. Let $\{f_n:n\in\mathbb N\}$ be a separable subset of $C(\mathbb R)$.
A function $f’:\mathbb N\to\mathbb R$ such that $f’(n)=f_n(n)+1$ is continuous on a discrete space $\mathbb N$. Tietze-Urysohn theorem implies that there exists a continuous extension $f:\mathbb R\to\mathbb R$ of the function $f’$.  Then $||f-f_n||\ge 1$ for each $n$, a contradiction.
