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 $[a, b]$.

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 Pinelis Jan 28 at 14:32

$\begingroup$ Sorry, what did you mean by density? $\endgroup$ – James Baxter Jan 28 at 14:34

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) LebesgueStieltjes measure $\mu_f$ is absolutely continuous with respect to the Lebesgue measure $\lambda$. What I mean then by the density is the RadonNikodym density $d\mu_f/d\lambda$. Can you answer the question about desired/expected terms of the characterization of the property? $\endgroup$ – Iosif Pinelis Jan 28 at 14:45

$\begingroup$ 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 Baxter Jan 28 at 14:48

$\begingroup$ 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 Baxter Jan 28 at 14:49
By Locally BV, $Vf(a,b)$ is finite. Moreover $Vf(a,b) \ge Vf(c,d)$,whenever $(c,d) \subset (a,b)$. So you are right, its finite always.
Edit : After I read the question correctly.
consider $f(x) = 0$ everywhere except at $x = x_0$, where $f(x_0) = a \in \mathbb{R}$. $f$ is locally BV. So you are wrong. The limit is not always finite.



$\begingroup$ $V(f)[a,b]$ is finite, sure. But this is not what the question is about. $\endgroup$ – Wojowu Jan 28 at 13:10

$\begingroup$ @Wojowu : thanks for pointing. I have edited the answer, after reading the question correctly. $\endgroup$ – user102868 Jan 28 at 13:15

$\begingroup$ You are again answering the wrong question. I presume OP knows the limit is not always finite, the question is under what condition it is. $\endgroup$ – Wojowu Jan 28 at 13:48