# Definition:Real Function

## 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 real numbers.

Hence the adjective **real** is often 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, some more vague and reliant upon intuition than others.

The transcluded pages are characteristic of the presentation of the subject which is not based on a treatment of set theory.

They are included for historical interest.

### Definition 1

Let $S \subseteq \R$ be a subset of the set of real numbers $\R$.

Suppose that, for each value of the independent variable $x$ of $S$, there exists a corresponding value of the dependent variable $y$.

Then the dependent variable $y$ is a **(real) function** the independent variable $x$.

### Definition 2

A **(real) function** is correspondence between a domain set $D$ and a range set $R$ that assigns to each element of $D$ a unique element of $R$.

### 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 $\map f x = y$.

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.

### Range

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

The range of $f$ is the set of values that the dependent variable can take.

## 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:

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

## Function of Two Variables

Let $S, T \subseteq \R$ be subsets of the set of real numbers $\R$.

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

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

The expression:

- $z = \map f {x, y}$

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:

- $y = \map f {x_1, x_2, \ldots, x_n}$

means:

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

## Examples

### Square Function

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

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

### Square Root Function

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

- $\forall x \in \R_{\ge 0}: \map f x = \sqrt x$

## Also see

- Definition:Mapping: the general definition

- 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 $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory - 1978: Garrett Birkhoff and Gian-Carlo Rota:
*Ordinary Differential Equations*(3rd ed.) ... (previous) ... (next): Chapter $1$ First-Order Differential Equations: $1$ Introduction: Example $1$ (footnote $\dagger$)