$\displaystyle \int_{0}^{1}{|f-g|}=0 $
but
$\displaystyle \sup_{x\in[0,1]}{|f-g|}>0 $
f,g are two continuous functions on [0,1]
You know that if $\displaystyle f\;\&\;g$ are continuous functions then $\displaystyle \left| {f - g} \right|$ is also a continuous function.
If a continuous function is positive at any point in $\displaystyle [0,1]$ then the function is positive throughout some subinterval.
What does that say about the integral?