1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Real analysis

  1. Nov 18, 2008 #1
    1. The problem statement, all variables and given/known data

    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]

    3. 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.
  2. jcsd
  3. Nov 18, 2008 #2
    you need to work the fact that f is continous 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.
  4. Nov 18, 2008 #3


    User Avatar
    Science Advisor
    Homework Helper

    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)?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook