# are there two continuous functions in [0,1] satisfies these conditions

• Nov 3rd 2008, 11:13 AM
szpengchao
are there two continuous functions in [0,1] satisfies these conditions
$\int_{0}^{1}{|f-g|}=0$
but
$\sup_{x\in[0,1]}{|f-g|}>0$

f,g are two continuous functions on [0,1]
• Nov 3rd 2008, 12:28 PM
Plato
You know that if $f\;\&\;g$ are continuous functions then $\left| {f - g} \right|$ is also a continuous function.
If a continuous function is positive at any point in $[0,1]$ then the function is positive throughout some subinterval.
What does that say about the integral?