# Separately continuous function

## Contents

## Definition at a point

### Generic definition

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

We say that a function 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 is a real-valued function of two variables , i.e., the domain of is a subset of . Suppose is a point in the domain of , i.e., it is the point where and (here are actual numerical values). We define three notions:

- is continuous with respect to at the point if the function (viewed as a function of one variable ) is continuous at .
- is continuous with respect to at the point if the function (viewed as a function of one variable ) is continuous at .
- is
*separately continuous*at the point if it is continuous with respect to*and*continuous with respect to at the point .

### For a function of multiple variables

Suppose is a real-valued function of variables , i.e., the domain of is a subset of . Suppose is a point in the domain of , i.e., it is the point where (here are actual numerical values). We define two notions:

- For each , we say that is continuous in at the point if the function is continuous at .
- We say that is
*separately continuous*in terms of*all*the inputs at a point if it is continuous with respect to at for each .

## Definition as a function on an open domain

### Generic definition

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

We say that 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 is a real-valued function of two variables , i.e., the domain of is an open subset of . Then:

- is continuous with respect to if is continuous with respect to at all points in its domain.
- is continuous with respect to if is continuous with respect to at all points in its domain.
- is separately continuous in if is continuous with respect to and continuous with respect to at all points in its domain.

### For a function of multiple variables

Suppose is a real-valued function of variables , i.e., the domain of is an open subset of :

- For each , we say that is continuous in if is continuous in for every point in its domain.
- We say that is
*separately continuous*in terms of*all*the inputs if it is continuous in all the inputs for every point in its domain.

## Graphical interpretation

### For a function of two variables

Suppose is a function of two variables . We consider the graph of as the subset in three-dimensional space with coordinate axes .

We have the following:

Assertion about continuity | How we can verify it from the graph |
---|---|

is continuous in at the point | Consider the graph restricted to the plane . This is continuous at . |

is continuous in at the point | Consider the graph restricted to the plane . This is continuous at . |

is separately continuous continuous in both variables at the point . | Both the above conditions. |

is continuous in everywhere. | The restrictions of the graph to all planes parallel to the -plane give graphs of continuous functions. |

is continuous in everywhere. | The restrictions of the graph to all planes parallel to the -plane give graphs of continuous functions. |

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 -plane or the -plane are graphs of continuous functions. |

### For a function of multiple variables

Suppose is a function of variables and a point is in the domain. Consider the graph of in given by . We have the following:

We have the following:

Assertion about continuity | How we can verify it from the graph |
---|---|

is continuous in at the point | Consider the graph restricted to the plane . This graph is continuous at . |

is separately continuous in all variables at the point . | The above holds for all . |

is continuous in everywhere. | The restrictions of the graph to all planes parallel to the -plane are continuous functions. |

is separately continuous in all variables everywhere. | The above holds for all . |