Interval notation

In this section, our functions will always have codomain R, and the domain will always be as much of R as possible (unless stated otherwise). So we'll often be dealing with subsets of the real line, especially connected
intervals such as fx : 2 x < 5g. It's a bit of a pain to write out the full set notation like this, but it sure beats having to say \all the numbers between 2 and 5, including 2 but not 5." We can do even better using interval notation.

We'll write [a; b] to mean the set of all numbers between a and b, including a and b themselves. So [a; b] means the set of all x such that a x b. For example, [2; 5] is the set of all real numbers between 2 and 5, including 2 and 5. (It's not just the set consisting of 2, 3, 4, and 5: don't forget that there are loads of fractions and irrational numbers between 2 and 5, such as 5=2,p7,and .) An interval such as [a; b] is called closed.

If you don't want the endpoints, change the square brackets to parentheses. In particular, (a; b) is the set of all numbers between a and b, not including a or b. So if x is in the interval (a; b), we know that a < x < b. The set (2; 5) includes all real numbers between 2 and 5, but not 2 or 5. An interval of the form (a; b) is called open.

You can mix and match: [a; b) consists of all numbers between a and b, including a but not b. And (a; b] includes b but not a. These intervals are closed at one end and open at the other. Sometimes such intervals are called half-open. An example is the set fx : 2 x < 5g from above, which can also be written as [2; 5).

There's also the useful notation (a; 1) for all the numbers greater than a not including a; [a; 1) is the same thing but with a included. There are three other possibilities which involve  1; all in all, the situation looks like this: