# Short question on functions of bounded variation

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 $$[a, b]$$.

• 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. – Iosif Pinelis Jan 28 at 14:32
• Sorry, what did you mean by density? – James Baxter Jan 28 at 14:34
• 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? – Iosif Pinelis Jan 28 at 14:45
• Hmm it’s hard to explain the full motivation, but I want something sort of like a weaker derivative for locally bounded variation functions. – James Baxter Jan 28 at 14:48
• 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? – James Baxter Jan 28 at 14:49

## 1 Answer

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.

• Right about what? – Wojowu Jan 28 at 12:25
• always finite. I mean about being finite. – user102868 Jan 28 at 12:56
• $V(f)[a,b]$ is finite, sure. But this is not what the question is about. – Wojowu Jan 28 at 13:10
• @Wojowu : thanks for pointing. I have edited the answer, after reading the question correctly. – user102868 Jan 28 at 13:15
• 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. – Wojowu Jan 28 at 13:48