About the region below the graph of a function

This paper deals with the hypograph of a function, i.e., the set of points lying on or below its graph. We will focus on a specific question: if and when a hypograph is connected. Special attention is devoted to the particular cases of additive discontinuous functions and everywhere surjections (constructions of everywhere surjections are also supplied). We give characterizations both of connected hypographs and of pathwise connected hypographs. In this context, simple examples arise of connected but not pathwise connected plane sets (in fact, with uncountably many path-components).