TITLE: Why are continuous functions not dense in $L^\infty$?
QUESTION [28 upvotes]: Why are the continuous functions not dense in $L^\infty$?
I mean both concretely (i.e. a counter example) and intuitively why is this the case.

REPLY [26 votes]: If a convergent sequence of continuous functions converges uniformly then the limiting function is continuous. Consequently, the only functions which can be approximated to arbitrary accuracy in $L^\infty$ by continuous functions are the continuous functions. In other words, the subspace of $L^\infty$ consisting of continuous functions is closed.
Of course, there are many functions in $L^\infty$ which are not continuous!