Points of increase of functions in the plane

Abstract

The set of continuous functions of a single nonnegative real variable with at least one point of increase is a “small” set under two natural definitions: on the one hand, it is a set of first Baire category, and on the other hand, according to a famous result of Dvoretsky, Erdös and Kakutani, it is also a set with Wiener measure zero. In this paper, an analogous question for continuous functions of two nonnegative real variables is examined. Consider the set of continuous functions \(f\) on the nonnegative quadrant for which there exists a monotone curve along which the restriction of \(f\) has a point of increase. In this paper, it is shown that this set is “small” in the sense that it is of first Baire category. However, this set is “large” in the sense that it has full measure under the probability measure induced by the standard Brownian sheet.