On a Lebesgue Density Property of Subsets of Real Numbers Related to Smital′s Lemma

Abstract

We will prove that if A, B are subsets of the real line R with positive outer Lebesgue measure and ƒ is a function of two real variables which is locally of class C1 at a point (a, b) ∈ A × B, where a, b are outer density points of A and B respectively and have nonvanishing partial derivatives at (a, b), then there exists a nonempty open interval I such that m*(ƒ(A × B) ∩ J) = m(J) for every nonempty open subinterval J of I. Here m, m* denote Lebesgue measure and outer Lebesgue measure respectively.