Some aspects of calculus on non-smooth sets

Let E be a closed set in R, and suppose that there is a k ≥ 1 such that every x, y ∈ E can be connected by a rectifiable path in E with length ≤ k |x−y|. This condition is satisfied by chord-arc curves, Lipschitz manifolds of any dimension, and fractals like Sierpinski gaskets and carpets. Note that length-minimizing paths in E are chord-arc curves with constant k. A basic feature of this condition is that one can integrate local Lipschitz conditions on E to get global conditions. For instance, if f : E → R is locally Lipschitz of order 1 with constant C ≥ 0, then f is globally Lipschitz on E with constant k C. Let A(x) be a continuous function on E with values in linear mappings from R to R. Equivalently, one can use the standard inner product on R to represent A(x) by a continuous mapping from E into R. Also let f be a locally Lipschitz real-valued function on E. Suppose that A includes the directional derivatives of f almost everywhere on any rectifiable curve in E, in the sense that the derivative of f(p(t)) is equal to A(p(t)) applied to the derivative of p(t) for almost every t when p(t) is a locally Lipschitz function on an interval I in the real line with values in E. In particular, this holds when f is continuously differentiable on E in the sense of the Whitney extension theorem with differential at x ∈ E given by A(x). In this case,