A Generalization of the Weierstrass Theorem

Abstract

The well-known Weierstrass theorem stating that a real-valued continuous function $f$ on a compact set $K\subset \mathbb{R}$ attains its maximum on $K$ is generalized. Namely, the space of real numbers is replaced by a set $Y$ with arbitrary preference relation $p$ (in place of the inequality ≤), and the assumption of continuity of $f$ is replaced by its monotonic semicontinuity (with respect to the relation $p$).