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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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Let f:J→R be an absolutely continuous and f'\in...?

Let $f:J\rightarrow \mathbb{R}$ be an absolutely continuous. Under what kind of extra condition for $f'$, (not $C$) holds the following relation? $$ \Big | \frac{1}{|I_{1}|}\int_{I_{1}}f'(x)dx- \f …
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