Separately continuous function

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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 \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 \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 \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 \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 \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_ix_{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\}.