Suppose that f is continuous function on the interval [a,b]

[e179285]

Suppose that f is continuous function on the interval [a,b]


integral from b to a If(x)I dx =0 if and only if f(x)=0 for all x in [a,b]

ıs it true or false ? ı can prove that if f is zero,integral is zero but ı can,'t do that if integral is zero f is zero

Regards

 

[HallsofIvy]

I assume that your If(x)I is supposed to be |f(x)|.

If f is not identically 0, there exist some point at which |f(x)| is positive. Since f is continuous, there is some interval on which |f(x)| is postive and so the integral over that interval is positive. And since |f(x)| is never negative there will not be any "canceling".