On the algebraic representation of semicontinuity

The concepts of upper and lower semicontinuity in pointfree topology were introduced and first studied by Li and Wang [Y.-M. Li, G.-J. Wang, Localic Katětov–Tong insertion theorem and localic Tietze extension theorem, Comment. Math. Univ. Carolin. 38 (1997) 801–814]. However Li and Wang’s treatment does not faithfully reflect the original classical notion. In this note, we present algebraic descriptions of upper and lower semicontinuous real functions, in terms of frame homomorphisms, that suggest the right alternative to the definitions of Li and Wang. This fixes the discrepancy between the classical and the pointfree notions and turns out to be the appropriate notion that makes the Katětov–Tong theorem provable in the pointfree context without any restrictions.