# Difference between Lebesgue theorem on Differentiability of functions of bounded variation and Rademacher theorem on 1D functions

I am currently trying to understand Rademachers theorem for $\mathbb{R}^N$ functions, and stumbled upon the following problem: On $\mathbb{R}$, to my knowledge, locally Lipschitz continuous functions are absolutely continuous and thus of bounded variation. From Lebesgue's Theorem then follows differentiability almost everywhere. Now Rademacher on one dimensional functions states the locally Lipschitz continuous functions are differentiable almost everywhere. Isn't that the same? Or did I mix something up? Any help would be greatly appreciated! :)

The point is that, like always $\mathbb{R}$ is a very small place to be. How can you define a.c. functions on $\mathbb{R}^n$? You may answer functions in $W^{1,1}$ and I would kind of agree. However the point is that on the line ac functions happen to be difference of increasing functions, which are themseves functions differentiable almost everywhere. Moreover I shold remark how a $metric$ derivative, which is given by Rademacher is deeply different from the $measure$ $theoretical$ one given by the theory of ac functions. While Rademacher theorem holds true even if you change the measure with something like $\delta_0$ nothing would change. On the other hand, if you change the measure with respect to which you are integrating, the set of ac functions should change. Indeed it is a non trivial result that $W^{1,\infty}$ is made of Lipschitz functions. I hope these short discussion is clear enogh.