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.