If a function $f:\mathbb{R}^N\to\mathbb{R}$ is of bounded variation (in modern sense or in Tonelli sense or according to any of the existing definitions), then can we say that, given any point $x\in \mathbb{R}^N$ and a unit vector $a \in \mathbb{R}^N$ can we say that the directional limits $\lim\_{\alpha\to 0+}f(x+\alpha a)$ and $\lim\_{\alpha\to 0-}f(x+\alpha a)$ where $\alpha \in \mathbb{R}$ exist always? 


PS : I am asking this question in MO as I couldn't find any such result on Wikipedia or in google search. Apologies if its not appropriate here and in case, move it to math.SE



Does directional limits along any given direction, always exist for a function of bounded variation?
No, consider the characteristic function $ f $ of a union $ A$ of infinitely many disjoint closed annuli centered on the origin in $\mathbb R^2$.
If the annuli have sufficiently fast-shrinking radii then the boundary of the set $ A$ has finite length. So $ f $ is of bounded variation (see [Caccioppoli set](http://en.m.wikipedia.org/wiki/Caccioppoli_set)). But the limits you mentioned do not exist at the origin. 


