Is 1/x Integrable for f>0 and xf(x) Tending to Zero?

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Discussion Overview

The discussion revolves around the integrability of the function 1/x when multiplied by a positive measurable function f(x) that tends to zero as x approaches zero. Participants explore the conditions under which the integral of f over a specified interval remains finite, particularly in the context of a homework question.

Discussion Character

  • Exploratory
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant proposes that if f>0 is a measurable function from [0,infinity) to itself and xf(x) tends to zero as x approaches zero, then the integral of f over [0,ε] is finite for some positive ε.
  • Another participant counters this by providing the example f(x)=1/(x ln(x)), suggesting it may not satisfy the integrability condition.
  • A different participant questions the definition of 1/x as a function from [0,infinity) to itself, raising concerns about its measurability.
  • One participant acknowledges a misunderstanding regarding the behavior of 1/(x ln(x)) as x approaches zero, initially claiming it tends to zero but later correcting themselves.
  • Another participant expresses confusion about the integrability of 1/(x ln(x)) and later resolves their own question.
  • A participant expresses bafflement at the arbitrary definition of 1/0 as 81, indicating a lack of clarity in the discussion.

Areas of Agreement / Disagreement

Participants exhibit disagreement regarding the integrability of specific functions and the definitions involved. The discussion remains unresolved, with multiple competing views and uncertainties present.

Contextual Notes

There are limitations in the assumptions made about the behavior of functions as x approaches zero, as well as the definitions of certain functions. The discussion does not resolve these mathematical uncertainties.

Palindrom
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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 \epsilon for which the integral of f over [0,\epsilon ] 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?
 
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No, take f(x)=1/(x ln(x)).
 
1/x isn't even a function from [0,infinity) to itself, never mind a measurable one.
 
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...
 
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Ok, why is 1/(xln(x)) not integrable?

Another stupid question, I answered myself... Thanks!
 
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What baffles me is that you define 1/0 to be 81...
 

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