Real-valued function

In mathematics, a real-valued function or real function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.

Many important function spaces are defined to consist of real functions.

Let X be an arbitrary set. Let ${\displaystyle {\mathcal {F}}(X,{\mathbb {R} })}$ denote the set of all functions from X to real numbers R. Because R is a field, ${\displaystyle {\mathcal {F}}(X,{\mathbb {R} })}$ may be turned into a vector space and a commutative algebra over reals by adding the appropriate structure:

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