Interval
An interval is a set containing all the real numbers located between any two specific real numbers on the number line. It is a property of the set of real numbers that between any two real numbers, there are infinitely many more. Thus, an interval is an infinite set. An interval may contain its endpoints, in which case it is called a closed interval. If it does not contain its endpoints, it is an open interval. Intervals that include one or the other of, but not both, endpoints are referred to as half-open or half-closed.
Notation
An interval can be shown using set notation. For instance, the interval that includes all the numbers between 0 and 1, including both endpoints, is written {x | 0 x 1}, and read "the set of all x such that 0 is less than or equal to x and x is less than or equal to 1." The same interval with the endpoints excluded is written {x | 0 < x < 1}, where the less than symbol (<) has replaced the less than or equal to symbol (). Replacing only one or the other of the greater than or equal to signs designates a half-open interval, such as {x | 0 x < 1}, which includes the endpoint 0 but not 1. A shorthand notation, specifying only the endpoints, is also used to designate intervals. In this notation, a square bracket is used to denote an included endpoint and a parenthesis is used to denote an excluded endpoint. For example, the closed interval {x | 0 x 1} is written [0,1], while the open interval {x | 0 < x < 1} is written (0,1). Appropriate combinations indicate half-open intervals such as [0,1) corresponding to {x | 0 x < 1}.
An interval may be extremely large, in that one of its endpoints may be designated as being infinitely large. For instance, the set of numbers greater than 1 may be referred to as the interval {x | -1 < x < }, or simply (-1,). Notice that when an endpoint is infinite, the interval is assumed to be open on that end. For example the half-open interval corresponding to the nonnegative real numbers is [0,), and the half-open interval corresponding to the nonpositive real numbers is (-,0].
Applications
There are a number of places where the concept of interval is useful. The solution to an inequality in one variable is usually one or more intervals. For example, the solution to 3x + 4 10 is the interval (-,2].
The interval concept is also useful in calculus. For instance, when a function is said to be continuous on an interval [a,b], it means that the graph of the function is unbroken, no points are missing, and no sudden jumps occur anywhere between x = a and x = b. The concept of interval is also useful in understanding and evaluating integrals. An integral is the area under a curve or graph of a function. An area must be bounded on all sides to be finite, so the area under a curve is taken to be bounded by the function on one side, the x-axis on one side and vertical lines corresponding to the endpoints of an interval on the other two sides.
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