Summary: An Lp
di¤erentiable nondi¤erentiable function
J. Marshall Ash
Abstract. 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 realvalued function f of a real variable is di¤erentiable at x if there is a real
number f0
(x) such that
jf (x + h) f (x) f0
(x) hj = o (h) as h ! 0:
Fix p 2 (0; 1). A function is di¤erentiable in the Lp
sense at x if there is a real
number f0
p (x) such that
f (x + h) f (x) f0
p (x) h p
