Interval (original) (raw)

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Interval

An interval is a connected portion of the real line. If the endpoints and are finite and are included, the interval is called closed and is denoted [a,b] . If the endpoints are not included, the interval is called open and denoted (a,b) . If one endpoint is included but not the other, the interval is denoted [a,b)](http://mathworld.wolfram.com/images/equations/Interval/Inline5.svg) or ![(a,b] and is called a half-closed (or half-open interval).

An interval [a,a] is called a degenerate interval.

If one of the endpoints is +/-infty , then the interval still contains all of its limit points, so [a,infty)](http://mathworld.wolfram.com/images/equations/Interval/Inline9.svg) and ![(-infty,b] are also closed intervals. Intervals involving infinity are also called rays or half-lines. If the finite point is included, it is a closed half-line or closed ray. If the finite point is not included, it is an open half-line or open ray.

The non-standard notation ![]a,b for an open interval and ![[a,b or ![]a,b]](http://mathworld.wolfram.com/images/equations/Interval/Inline13.svg) for a half-closed interval is sometimes also used.

A non-empty subset of is an interval iff, for all a,b in X and c in R , a<=c<=b implies c in X . If the empty set is considered to be an interval, then the following are equivalent: