# Convergence of the difference quotient of a BV function

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?

• What is the meaning of the quotient if N>1? – Pietro Majer Apr 13 '19 at 20:18
• But then converging to anything as ||h|| goes to zero, seems problematic even for N=1 and u(x)=x – Pietro Majer Apr 13 '19 at 21:31
• @PietroMajer Actually, I've edited the question now. – Riku Apr 13 '19 at 23:58
• Now the quotient depends on $y$, but the putative limit does not …. – LSpice Apr 14 '19 at 0:58
• 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 $t$? – Skeeve Apr 14 '19 at 6:42