# Definition:Real Function

## Contents

## Definition

A **real function** is a mapping or function whose domain and codomain are subsets of the set of real numbers $\R$.

It is frequently understood in many areas of mathematics that the domain and codomain of any function under discussion are of the set of set of real numbers so the adjective **real** is usually omitted unless it is an important point to stress.

Because the concept of a **function** has been around for a lot longer than that of a general **mapping**, there is a lot more terminology that has developed up round the subject.

### Independent Variable

Let $f: \R \to \R$ be a real function.

Let $\map f x = y$.

Then $x$ is referred to as an **independent variable**.

### Dependent Variable

Let $f: \R \to \R$ be a (real) function.

Let $f \left({x}\right) = y$.

Then $y$ is referred to as a **dependent variable**.

### Domain

Let $f: S \to \R$ be a real function.

The domain of $f$ is the set $S$.

It is frequently the case that $S$ is not explicitly specified. If this is so, then it is understood that the domain is to consist of *all* the values in $\R$ for which the function is defined.

This often needs to be determined as a separate exercise in itself, by investigating the nature of the function in question.

## Formula

A function $f: S \to T$ can be considered as a **formula** which tells us how to determine what the value of $y \in T$ is when we have selected a value for $x \in S$.

### As an Equation

It is often convenient to refer to an equation or formula as though it were a function.

What is meant is that the equation *defines* the function; that is, it specifies the rule by which we obtain the value of $y$ from a given $x$.

### Square Function

For example, let $x, y \in \R$.

The **(real) square function** is the real function $f: \R \to \R$ defined as:

- $\forall x \in \R: \map f x = x^2$

We may express this as $y = x^2$, and use this equation to *define* this function.

This may be conceived as:

- For each $x \in \R$, the number $y \in \R$ assigned to it is that which we get by squaring $x$.

Another useful notation is:

- $\forall x \in \R: x \mapsto x^2$

## Examples

### Square Function

The **(real) square function** is the real function $f: \R \to \R$ defined as:

- $\forall x \in \R: \map f x = x^2$

## Function of Two Variables

Let $f: S \times T \to \R$ be a mapping where $S, T \subseteq \R$.

Then $f$ is defined as a **(real) function of two (independent) variables**.

The expression:

- $z = f \left({x, y}\right)$

means:

- (The dependent variable) $z$ is a function of (the independent variables) $x$ and $y$.

## Function of n Variables

The concept can be extended to as many independent variables as required.

Let $f: S_1 \times S_2 \times \cdots \times S_n \to \R$ be a mapping where $S_1, S_2, \ldots, S_n \subseteq \R$.

Then $f$ is defined as a **(real) function of $n$ (independent) variables**.

The expression:

- $y = f \left({x_1, x_2, \ldots, x_n}\right)$

means:

- (The dependent variable) $y$ is a function of (the independent variables) $x_1, x_2, \ldots, x_n$.

## Also see

- Results about
**Real Functions**can be found here.

## Sources

- 1947: James M. Hyslop:
*Infinite Series*(3rd ed.) ... (previous) ... (next): Chapter $\text I$: Functions and Limits: $\S 2$: Functions - 1959: E.M. Patterson:
*Topology*(2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 9$. Functions - 1964: William K. Smith:
*Limits and Continuity*... (previous) ... (next): $\S 2.2$: Functions - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $1$: Algebraic Structures: $\S 1$: The Language of Set Theory - 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 1.3$: Functions and mappings. Images and preimages - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 7.1$ - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $8$: The System of the World: Calculus