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## Main Question or Discussion Point

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?

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?

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