 
Summary: TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 149, June, 1970
A CHARACTERIZATION OF THE PEANO DERIVATIVE(1)
BY
J. MARSHALL ASH
Abstract. For each choice of parameters {ai,bi}, i=O, 1., n+e, satisfying
certainsimpleconditions,the expression
n+e
lim h n 2 aif(x+bih)h*O _=0
yields a generalizednth derivative.A functionf has an nth Peano derivativeat x if
and only if all the membersof a certainsubfamilyof these nth derivativesexist at x.
TheresultholdsforthecorrespondingLIderivatives.A uniformitylemmain theproof
(Lemma2) may be of independentinterest.
Also, a new generalizedsecond derivativeis introducedwhich differentiatesmore
functions than the ordinary second derivative but fewer than the second Peano
derivative.
Introduction. There are severaldefinitions of the nth derivativeof a function of
a real variable in addition to the classical one. The most important perhaps is
that due to Peano: the functionf has at a point xo a derivative if there is a poly
