TITLE: $X$ metric separable then $C(X)$ separable
QUESTION [5 upvotes]: Is it true, that if $X$ is a separable metric space, then the space of all continuous functions on $X$ with the supremum metric is also separable?

REPLY [2 votes]: No. Consider triangle-shaped function
$$
\varphi(x)=\max(1-2|x|,0)
$$ 
then for each binary sequence $s:\mathbb{N}\to\{0,1\}$ we define
$$
f_s(x)=\sum\limits_{n=1}^\infty s(n)\varphi(x-n)
$$
One can show that $\{f_s:s\in\{0,1\}^\mathbb{N}\}$ is uncountable set of functions with the property
$$
s'\neq s''\implies \Vert f_{s'}-f_{s''}\Vert_{C(\mathbb{R})}\geq 1
$$
This implies that $C_b(\mathbb{R})$ is not separable.