calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

, and especially

multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather ...

, the mean of a function is loosely defined as the average value of the

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

over its domain. In one variable, the mean of a function ''f''(''x'') over the interval (''a'',''b'') is defined by :

\bar=\frac\int_a^bf(x)\,dx.

Recall that a defining property of the average value

\bar

of finitely many numbers

y_1, y_2, \dots, y_n

is that

n\bar = y_1 + y_2 + \cdots + y_n

. In other words,

\bar

is the ''constant'' value which when ''added'' to itself

n

times equals the result of adding the

n

terms

y_1, \dots, y_n

. By analogy, a defining property of the average value

\bar

of a function over the interval

,b /math> is that

: \int_a^b\bar\,dx = \int_a^bf(x)\,dx In other words, \bar is the ''constant'' value which when '' integrated'' over,b /math> equals the result of
integrating f(x) over,b /math>. But the integral of a constant \bar is just

: \int_a^b\bar\,dx = \barx\bigr, _a^b = \barb - \bara = (b - a)\bar See also the first mean value theorem for integration, which guarantees
that if f is

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...

then there exists a point

c \in (a, b)

such that :

\int_a^bf(x)\,dx = f(c)(b - a)

The point

f(c)

is called the mean value of

f(x)

,b /math>. So we write \bar = f(c) and rearrange the preceding equation to get the above definition.

In several variables, the mean over a

relatively compact In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (sin ...

domain ''U'' in a

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

is defined by :

\bar=\frac\int_U f.

This generalizes the arithmetic mean. On the other hand, it is also possible to generalize the geometric mean to functions by defining the geometric mean of ''f'' to be :

\exp\left(\frac{\hbox{Vol}(U)}\int_U \log f\right).

More generally, in measure theory and

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

, either sort of mean plays an important role. In this context, Jensen's inequality places sharp estimates on the relationship between these two different notions of the mean of a function. There is also a ''harmonic average'' of functions and a ''quadratic average'' (or ''root mean square'') of functions.