Continuity and Open and Closed Sets

We generalize the notion of open and closed intervals to open and closed sets in .

When we make definitions and discuss several important theorems for functions of a single variable, we need the notion of an open interval or a closed interval. A typical example of an open interval is , which represents the set of all such that , and an example of a closed interval is , which represents the set of all such that . We need analogous definitions for open and closed sets in .

A set is a collection of distinct objects.

Given a set , we say that is an element of if is one of the distinct objects in , and we write to denote this.

Given two sets and , we say that is a subset of if every element of is also an element of write to denote this.

With these ideas in mind, we now discuss special types of subsets.

(Open Balls)

We give these definitions in general, for when one is working in since they are really not all that different to define in than in .

An open ball in centered at with radius is the set of all points such that the distance between and is less than .

In an open ball is often called an open disk.

(Interior and Boundary Points) Suppose that .

• A point is an interior point of if there exists an open ball .

  

  Intuitively, is an interior point of if we can squeeze an entire open ball centered at within .

• A point is a boundary point of if all open balls centered at contain both points in and points not in .

  

• The boundary of is the set that consists of all of the boundary points of .

Suppose that .

Which of the following are elements of ?

Which of the following are interior points of ?

Which of the following are boundary points of ?

The boundary is

We can now generalize the notion of open and closed intervals from to open and closed sets in .

(Open and Closed Sets)

• A set is open if every point in is an interior point.

  

• A set is closed if it contains all of its boundary points.

  

Determine if the following sets are open, closed, or neither.

• The set is open closed neither open nor closed .
• The set is open closed neither open nor closed .
• The set is open closed neither open nor closed .

(Bounded and Unbounded)

• A set is bounded if there is an open ball such that

  Intuitively, this means that we can enclose all of the set within a large enough ball centered at the origin.

• A set that is not bounded is called unbounded.

Which of the following sets are bounded?

Let's now look at a few examples.

Consider the function .

• The domain of the function is the set of all for which , which we can write in set
• The point is an interior point a boundary point not an element of .
• The point is an interior point a boundary point not an element of .
• The domain is open closed neither open nor closed and bounded not bounded .

We've already found the domain of this function to be

This is the region bounded by the ellipse . Since the region includes the boundary (indicated by the use of ""), the set contains does not contain all of its boundary points and hence is closed. The region is bounded unbounded as a disk of radius , centered at the origin, contains .

Determine if the domain of is open, closed, or neither, and if it is bounded.

As we cannot divide by , we find the domain to be In other words, the domain is the set of all points not on the line . For your viewing pleasure, we have included a graph:

Note how we can draw an open disk around any point in the domain that lies entirely inside the domain, and also note how the only boundary points of the domain are the points on the line . We conclude the domain is an open set a closed set neither open nor closed set . Moreover, the set is bounded unbounded .

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Source: https://ximera.osu.edu/mooculus/calculusE/continuityOfFunctionsOfSeveralVariables/digInOpenAndClosedSetsE

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