For simplicity, we follow the rules: X denotes a non empty set, x denotes an element of X , f , g denote partial functions from X to R, S denotes a σ-field of subsets of X , F denotes a function from Q into S, p denotes a rational number, r denotes a real number, n, m denote natural numbers, and A, B denote elements of S. Let us consider X and let us consider f . We say that f is finite if and ...

This paper refers to the theory of semirings continuous numerical functions, which has been developed within functional algebra. The object investigation is CP(X) partial functions on topological spaces X with values in field R real numbers. subject study subalgebras CP(X). Some properties lattices A(X) all possible and A1(X) identity are considered. structure atoms preatoms clarified. allowed ...

In 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 a...