Extended real functions in pointfree topology

AbstractIn pointfree topology, a continuous real function on a frame L is a map L(R)→L from the frame of reals into L. The discussion of continuous real functions with possibly infinite values can be easily brought to pointfree topology by replacing the frame L(R) with the frame of extended reals L(R¯) (i.e. the pointfree counterpart of the extended real line R¯=R∪{±∞}). One can even deal with arbitrary (not necessarily continuous) extended real functions. The main purpose of this paper is to investigate the algebra of extended real functions on a frame. Our results make it possible to study the class D(L) of almost real functions. In particular, we show that for extremally disconnected L, D(L) becomes an order-complete archimedean f-ring with unit