Define F(x) = integral (from x to 1) f(adsbygoogle = window.adsbygoogle || []).push({});

take limF(x) as x goes to 0.

I would like to give an example of a function on (0,1] so that the above limit exists but if we replace f with |f| the limit does not exist.

So I came up with f= [(2^n)/(n)](-1)^(n+1) for x in (1/(2^n),1/[2^(n-1)]] (which geometrically) at these endpoints gives us the alternating harmonic series, that is the limit above with this function is the limit of the alternating harmonic series.

Is there a simpler example?

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# Example of f integrable but |f| not integrable

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