Characterization of the existence of semicontinuous weak utilities for binary relations

We characterize the existence of semicontinuous weak utilities in a general framework, where the axioms of transitivity and acyclicity are relaxed to that of consistency in the sense of Suzumura (Economica 43:381–390, 1976). This kind of representations allow us to transfer the problem of the existence of the $$𝒢𝒪𝒞\mathscr{H}𝒜$$ set of a binary relation to the easier problem of getting maxima of a real function. Finally, we show that the maxima of these representations correspond to the different levels of satiation that each of individual has (an individual reaches his or her level of satiation when an increase of consuming an alternative product/service brings no increase in utility).