Localic Real Functions: a General Setting

A [semi-]continuous real function of a frame (locale) L has up to now been understood as a frame homomorphism from the frame L(R) of reals into L [as a frame homomorphism (modulo some conditions) from certain subframes of L(R) into L]. Thus, these continuities involve different domains. It would be desirable if all these continuities were to have L(R) as a common domain. This paper demonstrates that this is possible if one replaces the codomain L by S(L) — the dual of the co-frame of all sublocales of L. This is a remarkable conception, for it eventually permits to have among other things the following: lower semicontinuous + upper semicontinuous = continuous. In this new environment we will have the same freedom in pointfree topology which so far was available only to the traditional topologists, for the lattice-ordered ring Frm(L(R),S(L)) may be viewed as the pointfree counterpart of the lattice-ordered ring RX with X a topological space. Notably, we now have the pointfree version of the concept of an arbitrary not necessarily continuous function on a topological space. Extended real functions on frames are considered too.