Interval

An interval is a subset of the real numbers R of one of the following types:
(i) (a,b) = {x: a<x<b}
(ii) [a,b] = {x: a≤x≤b}
(iii) [a,b) = {x: a≤x<b}
(iv) (a,b] = {x: a<x≤b}
(v) (a,∞) = {x: x>a}
(vi) [a,∞) = {x: x≥a}
(vii) (-∞,b) = {x: x<b}
(viii) (-∞,b] = {x: x≤b}
(ix) R itself, the set of all real numbers

Note that a square bracket [ or ] indicates that the number is included in the interval, while a round bracket ( or ) indicates that it is not.

Intervals of type (i),(v),(vii) and (ix) are called open intervals; the intervals (ii),(vi),(viii) and (ix) are called closed intervals; the intervals (i),(ii),(iii),(iv) are called bounded intervals; the intervals (v),(vi),(vii),(viii),(ix) are unbounded intervals.

The length of the bounded intervals (i),(ii),(iii),(iv) is b-a in each case. The total length of a sequence of intervals is the sum of the lengths of the intervals. No allowance is made for the intersection of the intervals. For instance, the total length of the sequence {(1,2),(1.5,2.5)} is 1+1=2, despite the fact that the union of the sequence is an interval of length 1.5.

Intervals may be characterized by the following property: a subset I of R is an interval if and only if whenever x and y are elements of I and z is such that x ≤ z ≤ y, then z is in I as well. Intervals are precisely the connected? subsets of R.

If f: R→R is a continuous function and I is an interval, then its image f(I) is also an interval. This is one formulation of the intermediate value theorem.

Intervals play an important role in the theory of integration, because they are the simplest sets whose "size" or "measure" or "length" is easy to define (see above). One then has to extend the concept of measure to more complicated sets.

Many theorems in analysis require that certain functions be continuous or differentiable on an open interval.