Prove f(x)=0 if integral of |f(x)| is zero

[Ara macao]

Prove that if f is continuous on [a,b] and

$$\int_a^b |f(x)|\,dx = 0$$

then f(x) = 0 for all x in [a,b].

so I'll have to use an epsilon delta proof by contradiction here. I'll have to assume that there exists a c such that f(c) != 0 and for all x = f(c)/2, there exists a delta such that |f(x)-f(c)|< epsilon for |x-c| < delta. and then I should make |f(x)| > epsilon /2. This would contradict the original hypothesis...

But I'm getting confused here...

Thanks!

 

[benorin]

Are you using Riemann integration or Lebesgue integration?

 

[Ara macao]

Riemann integration