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.

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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 $(x_0,y_0)$.

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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 $i\in \{1,2,3,\dots ,n\}$.

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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 $x_j=a_j,j\ne i$. 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 $i\in \{1,2,3,\dots ,n\}$.