Path derivatives and growth control

Abstract:

We show that a standard theorem relating the growth of a function on a measurable set to constraints on its Dini derivates extends to a class of generalized derivatives. More precisely, we show that if $\left | \bar {F}’_E \right | \leqslant M$ and $\left | \underline {F}’_E \right | \leqslant M$ on a measurable set $A$, then $\lambda (F(A)) \leqslant M\lambda (A)$. Here $\lambda$ denotes Lebesgue measure and ${\bar F’_E}$ and $\underline {F}’_E$ are the extreme derivates of $F$ relative to a system of paths which satisfy the Intersection Condition [1]. In particular, the result holds in the setting of unilateral differentiation, approximate differentiation, preponderant differentiation and qualitative differentiation.