Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Test for integrability

  1. Dec 30, 2006 #1
    It started out as an attempt to solve a HW question (which I also posted in the appropriate forum), but now I'm just curious as to the general case;

    Assume f>0 is a measurable function from [0,infinity) to itself. Then if xf(x) tends to zero as x tends to zero, there is a positive [tex]\epsilon[/tex] for which the integral of f over [tex][0,\epsilon ][/tex] is finite.

    This is following the intuition that while 1/x isn't integrable, multypling it by anything that tends to zero is.

    What do you say? True, not true?
    Last edited: Dec 30, 2006
  2. jcsd
  3. Dec 30, 2006 #2


    User Avatar
    Homework Helper

    No, take f(x)=1/(x ln(x)).
  4. Dec 30, 2006 #3

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    1/x isn't even a function from [0,infinity) to itself, never mind a measurable one.
  5. Dec 30, 2006 #4
    Ok, one by one:

    StatusX, 1/ln(x) doesn't tend to zero when x tends to zero, it tends to infinity... Edit: wait, it does, I'm an idiot.

    matt - It's almost everywhere defined, what's the problem? Define 1/0=81...
    Last edited: Dec 30, 2006
  6. Dec 30, 2006 #5
    Ok, why is 1/(xln(x)) not integrable?

    Another stupid question, I answered myself... Thanks!
    Last edited: Dec 30, 2006
  7. Dec 30, 2006 #6

    Gib Z

    User Avatar
    Homework Helper

    What baffles me is that you define 1/0 to be 81...
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook