Interval (mathematics)

intervalopen intervalclosed intervalintervalshalf-open intervalrangeclosed intervalsInterval notationinclusiveopen intervals
In mathematics, a (real) interval is a set of real numbers lying between two numbers, the extremities of the interval.wikipedia
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Integral

integrationintegral calculusdefinite integral
Real 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.
Given a function f of a real variable x and an interval

Lebesgue measure

Lebesgue measurableLebesguemeasure
The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure.
Given a subset, with the length of interval given by, the Lebesgue outer measure is defined as

Open set

openopen subsetopen sets
The open intervals are open sets of the real line in its standard topology, and form a base of the open sets.
In mathematics, particularly in topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

Bounded set

boundedunboundedbounded subset
Bounded intervals are bounded sets, in the sense that their diameter (which is equal to the absolute difference between the endpoints) is finite.
Therefore, a set of real numbers is bounded if it is contained in a finite interval.

General topology

point-set topologypoint set topologytopology
The open intervals are open sets of the real line in its standard topology, and form a base of the open sets.
The standard topology on R is generated by the open intervals.

Base (topology)

basisbaseweight
The open intervals are open sets of the real line in its standard topology, and form a base of the open sets.
For example, the collection of all open intervals in the real line forms a base for a topology on the real line because the intersection of any two open intervals is itself an open interval or empty.

Maxima and minima

maximumminimumlocal maximum
An interval is said to be left-open if and only if it contains no minimum (an element that is smaller than all other elements); right-open if it contains no maximum; and open if it has both properties.
An important example is a function whose domain is a closed (and bounded) interval of real numbers (see the graph above).

Real number

realrealsreal-valued
In mathematics, a (real) interval is a set of real numbers lying between two numbers, the extremities of the interval.
(The standard ordering ≤ of the real numbers is not a well-ordering since e.g. an open interval does not contain a least element in this ordering.) Again, the existence of such a well-ordering is purely theoretical, as it has not been explicitly described.

Total order

totally orderedtotally ordered setlinear order
Intervals are likewise defined on an arbitrary totally ordered set, such as integers or rational numbers.
For any totally ordered set X we can define the open intervals (a, b) = {x : a < x and x < b}, (−∞, b) = {x : x < b}, (a, ∞) = {x : a < x} and = X.

Closed set

closedclosed subsetclosed sets
These definitions are usually extended to include the empty set and to the (left- or right-) unbounded intervals, so that the closed intervals coincide with closed sets in that topology.

Intermediate value theorem

Bolzano's theoremBolzano–Cauchy theoremintermediate value property
This is one formulation of the intermediate value theorem.
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval [a, b], then it takes on any value between f(a) and f(b) at some point within the interval.

Convex set

convexconvex subsetconvexity
The intervals are also the convex subsets of \R.
A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set.

Continuous function

continuouscontinuitycontinuous map
It follows that the image of an interval by any continuous function is also an interval.

Ball (mathematics)

ballopen ballballs
If \R is viewed as a metric space, its open balls are the open bounded sets, and its closed balls are the closed bounded sets.
The ball is a bounded interval when

Positive real numbers

positive realspositive real axislogarithmic measure
For example, is the set of positive real numbers also written ℝ +.
If is an interval, then determines a measure on certain subsets of, corresponding to the pullback of the usual Lebesgue measure on the real numbers under the logarithm: it is the length on the logarithmic scale.

Array data type

arrayarraysmulti-dimensional array
The notation is used in some programming languages; in Pascal, for example, it is used to formally define a subrange type, most frequently used to specify lower and upper bounds of valid indices of an array.
Most of those languages also restrict each index to a finite interval of integers, that remains fixed throughout the lifetime of the array variable.

Complex number

complexreal partimaginary part
is often used to denote an ordered pair in set theory, the coordinates of a point or vector in analytic geometry and linear algebra, or (sometimes) a complex number in algebra.
The principal value of log is often taken by restricting the imaginary part to the interval.

Infinity

infiniteinfinitely
Even in the context of the ordinary reals, one may use an infinite endpoint to indicate that there is no bound in that direction.
The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R (see also Hilbert's paradox of the Grand Hotel).

Hyperrectangle

orthotopeboxes5-fusil
In higher dimensions, the Cartesian product of n intervals is bounded by an n-dimensional hypercube or hyperrectangle.
In geometry, an n-orthotope (also called a hyperrectangle or a box) is the generalization of a rectangle for higher dimensions, formally defined as the Cartesian product of intervals.

Metric space

metricmetric spacesmetric geometry
If \R is viewed as a metric space, its open balls are the open bounded sets, and its closed balls are the closed bounded sets.
Note that in the context of intervals in the space of real numbers and occasionally regions in a Euclidean space a bounded set is referred to as "a finite interval" or "finite region".

Interval graph

interval(proper) interval graphs
In graph theory, an interval graph is an undirected graph formed from a set of intervals on the real line,

Interval estimation

interval estimateintervalInterval (statistics)
In statistics, interval estimation is the use of sample data to calculate an interval of possible values of an unknown population parameter; this is in contrast to point estimation, which gives a single value.

Region (mathematics)

regionRegion (mathematical analysis)regions
Intervals can be associated with points of the plane and hence regions of intervals can be associated with regions of the plane.

Interval finite element

Interval FEMInterval finite element method
In this approach x is an interval number for which the equation