 
Summary: PROCEEDINGSOF THE
AMERICANMATHEMATICALSOCIETY
Volume121,Number3, July1994
CHARACTERIZATIONS AND GENERALIZATIONS OF CONTINUITY
J. M. ASH, J. COHEN,C. FREILING,L. GLUCK,E. RIEDERS,AND G. WANG
(Communicatedby AndrewBruckner)
ABSTRACT. The condition f(x + 2h)  2f(x + h) + f(x) = o(l) (as h * 0)
at each x is equivalentto continuityfor measurablefunctions. But thereis a
discontinuousfunction satisfying 2f(x + 2h)  f(x + h)  f(x) = o(l) at each
x. The question of which generalizedRiemann derivativesof order 0 char
acterizecontinuityis studied. In particular,a measurablefunction satisfying
EZI1if(x + fih) _ 0 mustbe a polynomial.Onthe otherhand,for anyRie
mannderivativeof order 0 and any p E [1, oo], generalized LP continuity
is equivalentto LP continuityalmosteverywhere.
INTRODUCTION
In Zygmund'sbook Trigonometricseries, the space of Lipschitz 1 functions
is defined by the relation Ig(x + h)  g(x)I = 0(h), and what is now called
the Zygmund space A* is defined by the relation Ig(x + 2h)  2g(x + h) +
g(x)l = 0(h). These spaces are different. However, Zygmundpoints out that
if 0 < a < 1, the conditions
