An L di¤erentiable non-di¤erentiable function

There is a a set E of positive Lebesgue measure and a function nowhere di¤erentiable on E which is di¤erentible in the Lp sense for every positive p at each point of E. For every p 2 (0;1] and every positive integer k there is a set E = E(k; p) of positive measure and a function which for every q < p has k Lq Peano derivatives at every point of E despite not having an Lp kth derivative at any point of E. A real-valued function f of a real variable is di¤erentiable at x if there is a real number f 0 (x) such that jf (x+ h) f (x) f 0 (x)hj = o (h) as h! 0: Fix p 2 (0;1). A function is di¤erentiable in the L sense at x if there is a real number f 0 p (x) such that f (x+ h) f (x) f 0 p (x)h p = o (h) as h! 0; where kg (h)kp = 1 h R h h jg (t)j p dt 1=p . We have an in…nite family of generalized …rst derivatives indexed by the parameter p: Most generalized derivatives are not equivalent to the ordinary derivative at a single point, but many are equivalent on an almost everywhere basis. For example, the symmetric derivative, de…ned by f 0 s (x) = limh!0 f(x+h) f(x h) 2h , is zero for the absolute value function at x = 0 even though that function is not di¤erentiable at x = 0, but this phenomenon which occurs at the single point x = 0 never occurs on a set of positive measure: there cannot exist a set of positive measure E and a function g so that g0 s (x) exists at all points of E and g 0 (x) exists at no points of E.[K, page 217] In this sense the symmetric derivative is equivalent to ordinary di¤erentiation. So a natural question to ask here is whether in this sense the various L derivatives are di¤erent from ordinary di¤erentiation and from one another. The point of this paper is to answer “yes”to this question. If p1 < p2 and f is L2 di¤erentiable at x, then f is L1 di¤erentiable at x; since by Holder’s inequality, f (x+ h) f (x) f 0 p2 (x)h p1 2 1 p1 1 p2 f (x+ h) f (x) f 0 p2 (x)h p2 = o (h) 2000 Mathematics Subject Classi…cation. Primary 26A27; secondary 26A24. Key words and phrases. Lp derivative, Peano derivative, Lp Peano derivative, super density. This research was partially supported by a grant from the Faculty and Development Program of the College of Liberal Arts and Sciences, DePaul University. This paper is in …nal form and no version of it will be submitted for publication elsewhere. 1