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Let f be continuous on [a,b] and differentiable almost everywhere on (a,b). Show that $$ \int_a^b f'(x)\,\mathrm{d}x=f(b)-f(a) $$ if and only if $$ \int_a^b\lim_{n\to\infty}\text{Diff}_{1/n}f=\lim_{n\to\infty}\int_a^b\text{Diff}_{1/n}f $$

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    What is $\mathrm{Diff}$?2012-11-23
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    martini:Assume f is continuous. Extend f to take the value f(b) on (b,b+1] and for 0$Diff_h f(x)=[f(x+h)-f(x)]/h$ for all x in [a,b]. – 2012-11-23

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