Consider a BV function $u \in BV(\mathbb{R}^N; \mathbb{R}^N)$. What can be said about the difference quotient $$ \frac{u(x+\epsilon y)u(x)}{\epsilon} $$ regarding its convergence as $\epsilon \to 0$? That is, in what sense does it converge to the derivative $Du = D^{a.c.} u + D^{sing} u$? Under what assumptions does the convergence hold almost everywhere?
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1$\begingroup$ What is the meaning of the quotient if N>1? $\endgroup$ – Pietro Majer Apr 13 '19 at 20:18

$\begingroup$ But then converging to anything as h goes to zero, seems problematic even for N=1 and u(x)=x $\endgroup$ – Pietro Majer Apr 13 '19 at 21:31

$\begingroup$ @PietroMajer Actually, I've edited the question now. $\endgroup$ – Riku Apr 13 '19 at 23:58

$\begingroup$ Now the quotient depends on $y$, but the putative limit does not …. $\endgroup$ – LSpice Apr 14 '19 at 0:58

$\begingroup$ Maybe you want to consider $\frac{u(x+ty)  u(x)}{t}$ for fixed $y\in \mathbb R^N$ and to study it as a function of $x$ depending on a parameter $t$? $\endgroup$ – Skeeve Apr 14 '19 at 6:42