Real-valued function

In other words, it is a function that assigns a real number to each member of its domain.

The σ-algebra of Borel sets is an important structure on real numbers.

Moreover, a set (family) of real-valued functions on X can actually define a σ-algebra on X generated by all preimages of all Borel sets (or of intervals only, it is not important).

This is the way how σ-algebras arise in (Kolmogorov's) probability theory, where real-valued functions on the sample space Ω are real-valued random variables.

The extreme value theorem states that for any real continuous function on a compact space its global maximum and minimum exist.

Convergent sequences also can be considered as real-valued continuous functions on a special topological space.

Continuous functions also form a vector space and an algebra as explained above in § Algebraic structure, and are a subclass of measurable functions because any topological space has the σ-algebra generated by open (or closed) sets.

Real numbers are used as the codomain to define smooth functions.

A measure on a set is a non-negative real-valued functional on a σ-algebra of subsets.

More precisely, whereas a function satisfying an appropriate summability condition defines an element of Lp space, in the opposite direction for any f ∈ Lp(X) and x ∈ X which is not an atom, the value f(x) is undefined.

Each of Lp spaces is a vector space and have a partial order, and there exists a pointwise multiplication of "functions" which changes p, namely For example, pointwise product of two L2 functions belongs to L1.