Proving f=0 on [a,b] with Continuous Nonnegative f

[CarmineCortez]

Homework Statement

Let f map [a,b]-->R be a continuous nonegative function. Suppose Integral f(x)dx from a to b = 0 show that f = 0 on [a,b]

The Attempt at a Solution

Just not sure if this is good or not..

so the lower sum <= 0 = integral f(x) dx

but the lower sum must be 0 since f is non negative.

now if f >0 then the lower sum >0 so this is a contradiction, f = 0.

 

[SNOOTCHIEBOOCHEE]

you need to work the fact that f is continuous in there.

If you consider a function i.e f(x)= 1 for rationals , 0 for irrationals; the integral of this is zero, the lower sum is zero, but f is in no way 0 throughout the entire interval.

 

[Dick]

Sort of. The idea is right. But you didn't use that f is continuous, and if f isn't continuous, it's not true. Show that if f is NOT equal to 0, then that implies the integral of f is greater than zero. Pick a point x where f(x)>0. Can you see how to use continuity at x to show you can put a positive area rectangle under f (like in a lower sum)?