Approximate Continuity and Differentiability with Respect to Density Degree, An Application to BV and Sobolev Functions

We define and discuss the pointwise notion of m-approximate continuity (differentiability) for functions f : A subset of R-n -> (R) over bar, with m >= n. The function f is m-approximately continuous (differentiable) at x is an element of A if and only if there exists E subset of A such that x is an element of E, lim(r -> 0+) r(-m)L(n) (B-r(x) \ E) = 0 and f vertical bar E is continuous (differentiable) at x. For m = n this notion coincides with the usual pointwise notion of approximate continuity (differentiability) with respect to L-n. The larger m, the more a function f that is m-approximately coninuous (differentiable) at x will resemble a function that is continuous (differentiable) at x. Among the applications, we provide in particular the following couple of results:If n >= 2 then the functions in BV are m(0)-approximately continuous at L-n-almost every point, with m(0) := n + 1 + 1/n-1,The functions in W-1,W-p, with p is an element of [1, n), are (n + p*)-approximately continuous at L-n-almost every point, where p* is the Sobolev conjugate of p, i.e., p* = np/(n - p).