Function between Spaces¶

Definition¶

Consider a function $f\colon X \to Y$ where $X$ is a set in an $n$-dimensional space (e.g., $\mathbb{R}^n$) and $Y$ is a set in a $k$-dimensional space (e.g., $\mathbb{R}^k$). We have the following terminologies:

• it is called a univariate function (also known as single variable function, function of one variable) if $n=1$$1$;
• it is called a bivariate function (also known as function of two variables) if $n=2$$2$;
• it is called a multivariable function (also known as multivariate function, function of several variables) if $n\geq 2$$2$;
• it is called a real-valued function if its codomain is a subset of $\mathbb{R}$;
• it is called a vector-valued function if $k\geq 2$$2$.

Interpretation¶

A function is classified by how many numbers go in and how many come out: $n$ counts the input variables and $k$ the output components. These names describe only the shape of the inputs and outputs; they say nothing about the rule itself.

The rows of the figure separate $n=1$ from $n\ge2$$2$, and the columns separate $k=1$ from $k\ge2$$2$. Each type is drawn in its own way. A univariate real-valued function has a graph in the plane. A function $\mathbb{R}\to\mathbb{R}^2$ is drawn as the curve traced by its outputs. A function $\mathbb{R}^2\to\mathbb{R}$ is drawn by its level sets. A function $\mathbb{R}^2\to\mathbb{R}^2$ is drawn as a vector field, with an output arrow attached at each input point. The graph of $f\colon\mathbb{R}^n\to\mathbb{R}^k$ lies in $\mathbb{R}^{n+k}$, so it can be drawn directly only when $n+k\le3$$3$.

A vector-valued function $\boldsymbol{f}=(f_1,\dots,f_k)$ is the same as $k$ real-valued functions of the same input; it is written in bold because its values are vectors. For example, $\boldsymbol{f}(t)=(\cos t,\sin t)$ from $\mathbb{R}$ to $\mathbb{R}^2$ traces the unit circle, and each component is an ordinary function of one variable from Math I.

A common slip is to mix up the two counts. "Multivariable" refers to the inputs and "vector-valued" to the outputs: $f(x,y)=x^2+y^2$ is multivariable but real-valued, and $\boldsymbol{f}(t)=(\cos t,\sin t)$ is vector-valued but univariate.

This idea returns later: the derivative of $\boldsymbol{f}\colon\mathbb{R}^n\to\mathbb{R}^k$ is a $k\times n$ matrix, the Jacobian matrix, with one row per output and one column per input.

Key Results¶

• A multivalued function (a rule that may assign several outputs to one input, such as $x\mapsto\pm\sqrt{x}$$\pm \sqrt{x}$) is not a function in the sense used here, so it is neither a multivariable function nor a vector-valued function.

Examples¶

• $f(x,y) = x^2 + y^2$ from $\mathbb{R}^2$ to $\mathbb{R}$ is a bivariate real valued function.
• $\boldsymbol{F}(x,y) = (x+y, x-y, xy)$ from $\mathbb{R}^2$ to $\mathbb{R}^3$ is a multivariable vector valued function.

Prerequisites¶

• Function

Related Concepts¶

• Random Variable

Sources¶

• Math III: Analysis · Lecture 7

Aliases: multivariable function, multivariate function, function of several variables Symbols: real numbers, square root

Last updated: 20 October 2026 03:02