All functions here are from R to R and I use the supremum metric d.
$C_c$= continuous functions with compact support and $C_b$ = continuous bounded functions.
We say a metric space $(X,d)$ is of first category if it can be written as a countable union of nowhere dense (rare) subsets. Otherwise, it is of second category.
a) What is a category of $C_c$ in $C_b?$
b) What is a category of $C_c$ in itself?
As far as I know, $(C_b,d)$ is complete, so we can also use Baire's category theorem. I already show that the completion of $C_c$ is $C_0$; continuous functions vanishing at infinity. I guess this implies that $C_c$ cannot be dense in $C_b$, so maybe this fact might be useful for part a). To be honest, for part b) I have no idea.