Functions of the Real Numbers

Real-Valued Functions

Definition: Real-Valued Function

A real-valued function is a function whose codomain are the real numbers.

Definition: Operations with Real-Valued Functions

Let $f:{\mathcal{D}}_f\to \mathbb{R}$ and $g:{\mathcal{D}}_g\to \mathbb{R}$ be real-valued functions.

We define

• the sum of $f$ and $g$ as $f+g:{\mathcal{D}}_f\cap {\mathcal{D}}_g\to \mathbb{R}$ with $(f+g)(x)=f(x)+g(x)$;
• the difference $f-g:{\mathcal{D}}_f\cap {\mathcal{D}}_g\to \mathbb{R}$ with $(f-g)(x)=f(x)-g(x)$;
• the product of $f$ and $g$ as $fg:{\mathcal{D}}_f\cap {\mathcal{D}}_g\to \mathbb{R}$ with $(fg)(x)=f(x)g(x)$;
• the quotient $f/g:\mathcal{D}\to \mathbb{R}$ with $\mathcal{D}=\{x\in {\mathcal{D}}_f\cap {\mathcal{D}}_g\mid g(x)\ne 0\}$ and $(f/g)(x)=\frac{f(x)}{g(x)}$.

Real Functions

Definition: Real Function

A real function is a real-valued function whose domain is a subset of the real numbers:

$$f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$$

Real Vector-Valued Functions

Definition: Real Vector-Valued Function

A real vector-valued function $f:D\to {\mathbb{R}}^n$ is a function from an arbitrary set $D$ to the real vector space ${\mathbb{R}}^n$.

Note: Component Functions

Every real vector-valued function $f:D\to {\mathbb{R}}^n$ can be described by $n$ real-valued functions $f_1,\dots ,f_n:\mathcal{D}\to {\mathbb{R}}^n$, where $f_k$ is responsible for the $k$-th component of the resulting vector:

$$f(x)=\left[\begin{array}{c}{f}_1(x)\\ ⋮\\ f_n(x)\end{array}\right]$$

Hence, $f_1,\dots ,f_n$ are called the component functions of $f$.

Real Vector Functions

Definition: Vector Function

A vector function $f:\mathcal{D}\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ is a real vector-valued function whose domain $\mathcal{D}$ is a subset of a real vector space ${\mathbb{R}}^m$.

Notation: Multivariate Notation and Coordinate Representations

Strictly speaking, a Real Vector Function $f$ takes a vector $\mathbf{p}\in \mathcal{D}\subseteq {\mathbb{R}}^m$ and outputs another vector $f(\mathbf{p})\in {\mathbb{R}}^n$. However, since vectors live as a points in Euclidean space, they can be uniquely represented by coordinates.

Although the action of $f$ is independent of the choice of coordinate systems for ${\mathbb{R}}^m$ and ${\mathbb{R}}^n$ ($\mathbf{p}$ and $f(\mathbf{p})$ are the same vectors, regardless of how we choose to represent them), many definitions involving $f$, such as differentiability and integrability, do depend on the choice of a coordinate system for the input space ${\mathbb{R}}^m$. Indeed, it would be more appropriate to formulate such definitions about the coordinate representations of $f$ in the chosen coordinate systems. However, this formality gets very cumbersome very quickly.

Hence, if the coordinates of $\mathbf{p}$ in a given coordinate system $\varphi$ are $p^1,\dots ,p^n$, then instead of writing ${}_{({\mathbb{R}}^m,\varphi )}f(p^1,\dots ,p^m)$ for the coordinate representation of $f$, we can just write $f(p^1,\dots ,p^m)$, as long as it is clear which coordinate system we are using. If there is no particular coordinate system specified, then we usually assume that those are Cartesian coordinates.

This is why vector functions are often called multivariate functions.

Note: Component Functions

The component functions of $f$ are real scalar fields.