Graph‐null sets

Abstract

We say that a plane set is graph‐null , if there is a function such that . A plane set has the translational Kakeya property if, for every translated copy of and for every , there is a finite sequence of vertical and horizontal translations bringing to such that the area touched during the horizontal translations is less than . These properties are equivalent if is compact. We show that the graph of every absolutely continuous function is graph‐null. Also, the graph of a typical continuous function is graph‐null. Therefore, there are nowhere differentiable continuous functions whose graphs are graph‐null. Still, we show that there exists a continuous function whose graph is not graph‐null.