Does directional limits along any given direction, always exist for a function of bounded variation?

Question

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

Answer 1

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). But the limits you mentioned do not exist at the origin.