For a function $f: \mathbb R \to \mathbb R$ of locally bounded variation, when is $$\liminf_{e \to 0} V(f)[x, x+e]/e $$finite everywhere? Here $V(f)[a, b]$ denotes the total variation of the function on the interval $[a, b]$.
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1$\begingroup$ An obvious sufficient condition for this property is that $f$ be absolutely continuous with a locally bounded density. As for a necessary and sufficient condition, it is unclear in what terms you want such a characterization to be. I think it's possible that there are no (more or less) simple characterizations of this property, except for the trivial tautology: the property holds iff it holds. This property reminds me a bit the strong law of large numbers for not necessarily independent identically distributed random variables, for which I think no general nontrivial characterization exists. $\endgroup$– Iosif PinelisCommented Jan 28, 2019 at 14:32
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$\begingroup$ Sorry, what did you mean by density? $\endgroup$– James BaxterCommented Jan 28, 2019 at 14:34
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1$\begingroup$ By one of equivalent definitions of absolute continuity (en.wikipedia.org/wiki/Absolute_continuity), if $f$ is absolutely continuous, then the corresponding (signed) Lebesgue--Stieltjes measure $\mu_f$ is absolutely continuous with respect to the Lebesgue measure $\lambda$. What I mean then by the density is the Radon--Nikodym density $d\mu_f/d\lambda$. Can you answer the question about desired/expected terms of the characterization of the property? $\endgroup$– Iosif PinelisCommented Jan 28, 2019 at 14:45
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$\begingroup$ Hmm it’s hard to explain the full motivation, but I want something sort of like a weaker derivative for locally bounded variation functions. $\endgroup$– James BaxterCommented Jan 28, 2019 at 14:48
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$\begingroup$ Because requiring a derivstive to exist at every point would be a little too extreme for the class of BV functions, I want some kind of “rough” measure of the pointwise change in the function, if that makes sense? $\endgroup$– James BaxterCommented Jan 28, 2019 at 14:49
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