Interval

An interval is a subset S of the real numbers R with the property that if x and y are in S and x < z < y then z is in S.

Every interval is of one of the following eleven types (where a and b are real numbers, with a < b):
(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
(x) {a}
(xi) the empty set

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), (ix) and (xi) are called open intervals; the intervals (ii), (vi), (viii), (ix), (x) and (xi) are called closed intervals; the intervals (i), (ii), (iii), (iv), (x) and (xi) 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 are precisely the connected subsets of R. Since a continuous image of a connected set is connected, it follows that 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.

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