Why are continuous functions not dense in $L^\infty$?

Question

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.

Answer 1

Consider $$f(x)=\begin{cases}0&\text{if }x<0\\1&\text{if }x\ge 0\end{cases}$$ Any continuous $g$ with $||f-g||_\infty<\frac 13$ must have $g(x)<f(x)+\frac13=\frac13$ for all $x<0$. By countinuity, $g(0)\le \frac13$, contradicting $g(0)>f(0)-\frac13=\frac 23$.

Even if we only require $|f(x)-g(x)|<\frac13$ for almost all $x$, the argument above still holds (with using continuity on the right as well).

Intuitively, the continuous $g$ cannot do the jump at once, it needs some "preparation" and "relaxation".