Existence of a functional on $L^\infty$ that vanish on C(X)

Question

So I want to prove the existence of a functional on $L^\infty(0,1)$ that vanish on C(X). I looked up and there is one example, but not much explanation. So we consider the continuous functions that have equal limits when x tends to 0 and 1. And take the functional that maps a function to the difference of two limits. And then extend it to the whole space.

However, it only vanishes on the set of continuous functions that have equal limits. Can you help me explain why does it vanishes on all continuous ones? I think this is wrong, do you know other examples?

Answer 1

Let $V$ be the span of $C[0,1]$ and some particular member of $L^\infty(0,1)$ that is not in $C[0,1]$, e.g. $I_{[0,1/2]}$, the indicator function of $[0, 1/2]$. Define a bounded linear functional $\varphi$ on $V$ by $\varphi(f + t I_{[0,1/2]}) = t$. Then extend to $L^\infty(0,1)$ by Hahn-Banach.