On the space of continuous functions of a second-countable space

Question

Let $X$ be a locally compact, Hausdorff, second-countable space, and let $C(X,\mathbb{C})$ the space of continuous, complex-valued functions over X endowed with the uniform norm.

Is $C(X,\mathbb{C})$ separable?

Answer 1

Not in general. Consider $X = \mathbb{R}$, for each subset $A$ of the integers let $f_A : \mathbb{R} \to \mathbb{C}$ be the function which has a narrow triangular spike (say of width 1/2) of height 1 centered at each integer in $A$, and is zero otherwise. The distance between any two such functions in the sup norm is one, and there are uncountably many of them, so the space cannot be separable.