Separately continuous function

Definition at a point

Generic definition

Suppose $f$ is a function of more than one variable, where $x$ is one of the input variables to $f$ . Fix a choice $x=x_{0}$ and fix values of all the other input variables. We say that $f$ is continuous with respect to $x$ at this point in its domain if the following holds: the function that sends $x$ to $f$ evaluated at $x$ and the fixed choice of the other input variables is continuous at $x=x_{0}$ .

We say that a function $f$ of several variables is separately continuous in the variables at a point if it is separately continuous with respect to each of the variables at the point.

For a function of two variables

Suppose $f$ is a real-valued function of two variables $x,y$ , i.e., the domain of $f$ is a subset of $\mathbb {R} ^{2}$ . Suppose $(x_{0},y_{0})$ is a point in the domain of $f$ , i.e., it is the point where $x=x_{0}$ and $y=y_{0}$ (here $x_{0},y_{0}$ are actual numerical values). We define three notions:

$f$ is continuous with respect to $x$ at the point $(x_{0},y_{0})$ if the function $x\mapsto f(x,y_{0})$ (viewed as a function of one variable $x$ ) is continuous at $x=x_{0}$ .
$f$ is continuous with respect to $y$ at the point $(x_{0},y_{0})$ if the function $y\mapsto f(x_{0},y)$ (viewed as a function of one variable $y$ ) is continuous at $y=y_{0}$ .
$f$ is separately continuous at the point $(x_{0},y_{0})$ if it is continuous with respect to $x$ and continuous with respect to $y$ at the point $(x_{0},y_{0})$ .

For a function of multiple variables

Suppose $f$ is a real-valued function of variables $x_{1},x_{2},\dots ,x_{n}$ , i.e., the domain of $f$ is a subset of $\mathbb {R} ^{n}$ . Suppose $(a_{1},a_{2},\dots ,a_{n})$ is a point in the domain of $f$ , i.e., it is the point where $x_{1}=a_{1},x_{2}=a_{2},\dots ,x_{n}=a_{n}$ (here $a_{1},a_{2},\dots ,a_{n}$ are actual numerical values). We define two notions:

For each $i\in \{1,2,3,\dots ,n\}$ , we say that $f$ is continuous in $x_{i}$ at the point $(a_{1},a_{2},\dots ,a_{n})$ if the function $x_{i}\mapsto f(a_{1},a_{2},\dots ,a_{i-1},x_{i},a_{i+1},\dots ,a_{n})$ is continuous at $x_{i}=a_{i}$ .
We say that $f$ is separately continuous in terms of all the inputs $x_{1},x_{2},\dots ,x_{n}$ at a point $(a_{1},a_{2},\dots ,a_{n})$ if it is continuous with respect to $x_{i}$ at $(a_{1},a_{2},\dots ,a_{n})$ for each $i\in \{1,2,3,\dots ,n\}$ .

Definition as a function on an open domain

Generic definition

Suppose $f$ is a function of more than one variable whose domain is open (i.e., has no boundary points in it). Suppose $x$ is one of the inputs to $f$ . We say that $f$ is continuous with respect to $x$ if it is continuous with respect to $x$ at all points in its domain.

We say that $f$ is separately continuous if it is continuous with respect to each of the variables that are inputs to it.

For a function of two variables

Suppose $f$ is a real-valued function of two variables $x,y$ , i.e., the domain of $f$ is an open subset of $\mathbb {R} ^{2}$ . Then:

$f$ is continuous with respect to $x$ if $f$ is continuous with respect to $x$ at all points in its domain.
$f$ is continuous with respect to $y$ if $f$ is continuous with respect to $y$ at all points in its domain.
$f$ is separately continuous in $x,y$ if $f$ is continuous with respect to $x$ and continuous with respect to $y$ at all points in its domain.

For a function of multiple variables

Suppose $f$ is a real-valued function of variables $x_{1},x_{2},\dots ,x_{n}$ , i.e., the domain of $f$ is an open subset of $\mathbb {R} ^{n}$ :

For each $i\in \{1,2,3,\dots ,n\}$ , we say that $f$ is continuous in $x_{i}$ if $f$ is continuous in $x_{i}$ for every point in its domain.
We say that $f$ is separately continuous in terms of all the inputs $x_{1},x_{2},\dots ,x_{n}$ if it is continuous in all the inputs for every point in its domain.

Graphical interpretation

For a function of two variables

Suppose $f$ is a function of two variables $x,y$ . We consider the graph of $f$ as the subset $z=f(x,y)$ in three-dimensional space with coordinate axes $x,y,z$ .

We have the following:

Assertion about continuity	How we can verify it from the graph
$f$ is continuous in $x$ at the point $(x_{0},y_{0})$	Consider the graph restricted to the plane $y=y_{0}$ . This is continuous at $x=x_{0}$ .
$f$ is continuous in $y$ at the point $(x_{0},y_{0})$	Consider the graph restricted to the plane $x=x_{0}$ . This is continuous at $y=y_{0}$ .
$f$ is separately continuous continuous in both variables at the point $(x_{0},y_{0})$ .	Both the above conditions.
$f$ is continuous in $x$ everywhere.	The restrictions of the graph to all planes parallel to the $xz$ -plane give graphs of continuous functions.
$f$ is continuous in $y$ everywhere.	The restrictions of the graph to all planes parallel to the $yz$ -plane give graphs of continuous functions.
$f$ is separately continuous in both variables everywhere.	Both the above conditions, i.e., the restrictions of the graph to all planes parallel to either the $xz$ -plane or the $yz$ -plane are graphs of continuous functions.

For a function of multiple variables

Suppose $f$ is a function of variables $x_{1},x_{2},\dots ,x_{n}$ and a point $(a_{1},a_{2},\dots ,a_{n})$ is in the domain. Consider the graph of $f$ in $\mathbb {R} ^{n+1}$ given by $x_{n+1}=f(x_{1},x_{2},\dots ,x_{n})$ . We have the following:

We have the following:

Assertion about continuity	How we can verify it from the graph
$f$ is continuous in $x_{i}$ at the point $(a_{1},a_{2},\dots ,a_{n})$	Consider the graph restricted to the plane $x_{j}=a_{j},j\neq i$ . This graph is continuous at $x_{i}=a_{i}$ .
$f$ is separately continuous in all variables at the point $(a_{1},a_{2},\dots ,a_{n})$ .	The above holds for all $i\in \{1,2,3,\dots ,n\}$ .
$f$ is continuous in $x_{i}$ everywhere.	The restrictions of the graph to all planes parallel to the $x_{i}x_{n+1}$ -plane are continuous functions.
$f$ is separately continuous in all variables everywhere.	The above holds for all $i\in \{1,2,3,\dots ,n\}$ .