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: Counterexamples needed for integration question

  1. Aug 5, 2007 #1
    1. The problem statement, all variables and given/known data

    The original question required me to show that for f(x) >= 0 for all x, f continuous, where the integral (from a to b) of f =0, that f(x) = 0 for all x in [a,b]. I did that, using a proof by contradiction.

    Second part of the question requires me to show that the two hypotheses (f(x) being >= 0 and f being continuous) were required.

    2. Relevant equations

    3. The attempt at a solution

    I think counterexamples would show this, but can't figure out what would make a counterexample. Do I need to take f(x)<0 for f continuous and show that its integral can't equal zero? Similarly, take f(x)>=0 but not continuous and show that its integral also can't equal zero?
  2. jcsd
  3. Aug 5, 2007 #2


    User Avatar
    Science Advisor
    Homework Helper

    Almost. You want the integral to be zero. And for the first bit, you don't really need f(x)<0 for all x, just f(x0)<0 for some x0 in [a,b]. (Actually if f(x)<0 for all x, it won't work, because the integral won't be zero.)
    Last edited: Aug 6, 2007
  4. Aug 5, 2007 #3
    so would taking f(x) = x^3, which is continuous, be a suitable counterexample for the first assumption?

    I can't think of a function that is always positive and isn't continuous, though.
  5. Aug 5, 2007 #4


    User Avatar
    Science Advisor
    Homework Helper

    Yup. That works provided you integrate on some interval symmetric about 0, like [-1,1].

    How about

    0 &\text{if } x \neq 0\\
    1 &\text{if } x = 0

    for the second one?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook