Calculus: Interval

An interval is nothing but an ordered set. It is defined in terms of 2 values a and b (a < b), such that all x, a <= x <= b, are in the interval. The points a and b are called the endpoints of the interval. All other points x are called the interior points. The set of all interior points is called the interior of the interval.

There are different types of intervals.

If both a and b are finite values, the interval is a bounded interval. Otherwise, it is an unbounded interval.

Further, there are what are called open and closed intervals depending on whether the interval "contains" the endpoints or boundaries. A closed interval, denoted as [a, b], is one that contains all its endpoints. Here, a and b are the boundaries of the interval. An open interval, denoted as (a, b), is one that does not contain its endpoints. Or it does not have boundary points.

This will be clearer if we associate intervals with functions. Now, a function is defined over an interval. If a function is defined over a closed interval [a, b], then the function has a value at each endpoint. That is, the function is defined everywhere in the interval. OTOH, in case of an open interval (a, b), the function is not defined at the endpoints.

Similarly, we have half open or half closed intervals. Example: [0, 20) or (-10, 55]. In the first case, a function is defined at every point from 0 (inclusive) up to 20 (exclusive).

The concept of intervals is very important in optimization.