Localic real functions: A general setting

In pointfree topology the lattice-ordered ring of all continuous real functions on a frame LL has not   been a part of the lattice of all lower (or upper) semicontinuous real functions on LL just because all those continuities involve different domains. This paper demonstrates a framework in which all those continuous and semicontinuous functions arise (up to isomorphism) as members of the lattice-ordered ring of all   frame homomorphisms from the frame L(R)L(R) of reals into S(L)S(L), the dual of the co-frame of all sublocales of LL. The lattice-ordered ring Frm(L(R),S(L)) is a pointfree counterpart of the ring RXRX with XX a topological space. We thus have a pointfree analogue of the concept of an arbitrarynot necessarily (semi) continuous   real function on LL. One feature of this remarkable conception is that one eventually has: lower semicontinuous + upper semicontinuous = continuous. We document its importance by showing how nicely can the insertion, extension and regularization theorems, proved earlier by these authors, be recast in the new setting.