This report is motivated by a study of Rogers and Taylor characteriz-
ing those interval functions which are absolutely continuous with respect to
the s-dimensional Hausdorff measure. This problem leads naturally to an
investigation of Lipschitz numbers
D(f,x)= limsup
y,z-^x,yxz \Z
and to s-dimensional integrals roughly of the form
= lim£/(£)(*.• - *i-i)*-
Ja i=i
The exposition is presented in the setting of interval functions on the real
line and the differentiation, measure-theoretic and variational properties are
developed. Applications are given to the Hausdorff and packing measures as
well as to the classical differentiation theory of real functions.
Received by the editor October 5, 1988. Received in revised form November 5,
Mathematics Subject Classification (1985 Revision). Primary 26A21, 26A24.
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