 
Summary: GENERALIZATIONSOF THE
RIEMANN DERIVATIVE(1)
BY
J. MARSHALL ASH
Introduction. In ?1 of this paper a derivative generalizingthe Riemann deriva
tive is considered. The existence of this derivative on a set is shown to imply the
existence of the Peano derivativealmost everywhereon the set. In ?2 the LI norm
(1 ?p
case of this result is that the existence of the Riemann LI derivative implies the
existence of the Peano LI derivative almost everywhere.In ?3 a generalization of
smoothness is shown to imply smoothness almost everywhere.We consider only
measurablesets of real numbers and real valued functions of a real variable.
1. An LX generalizationof the Riemannderivative. A functionf is said to have
a Peano derivative of order k at x, i.e., f E tk(X), if there are constants fo(x),
fA(x),... , fk(x) such that
f(x +t) = fo(x) +fi(x)t + ***+ k!
tk
+ (tk) as t O*.
We sayf is Peano bounded of order k at x, i.e., f E Tk(x), if there are constants
fo(x),.. ,fk_l(x) such that
