Suppose $f:\mathbb R^{n}\to \mathbb R$ is a Lipschitz function. Is $\sqrt{1+|\nabla f|^2}$ Riemann (not Lebesgue) integrable on a bounded open set, say a ball?
In $\mathbb R^1$, a function is Riemann integrable on a bounded interval $[a,b]$ iff it is continuous almost everywhere with respect to Lebesgue measure. Do we have an analogue for higher dimension?