A closed interval, as opposed to an open interval, is a straight lineconnectedOfClosed set, is the set of all points on the line between fixed two points (including the given two points), which is represented by [a, b] (including two endpoints a and b) (and a<b).Because it isBoundedClosed set, so it is compact.
Closed interval is a mathematical term, opposite to open interval.
Of all points on a line between fixed two pointsaggregate(including the given two points).The closed interval is on a straight lineconnectedOfClosed set。Because it isBoundedClosed set, so it is compact.
Representing symbols: [x, y], that is, from the value of x to the value of y, including x and y. For example, if the value range of x is a closed interval from 3 to 5, then it is expressed in mathematical language as [3,5], that is, the number from 3 (inclusive) to 5 (inclusive).
section
Announce
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sectionIt is the most commonly used point set on the number axis.There are three types of intervals:
1. Closed interval
, where a and b are real numbers (the same below);
2. Open interval
；
3. Semi open interval
Semi open interval is also called semi closed interval.The above a and b are called the left and right endpoints of the corresponding interval, and other points in the interval are called the interior points of the interval.
In the above sections,
Also called bounded interval or finite interval, other points are called unbounded interval or infinite interval.about
, interval
Also known as symmetric interval, interval is a point line segment or ray on the number axis or the whole number axis, and "open" ("closed", "semi open") means not including (including, only including one) its endpoint. In the expanded real number system R *, four kinds of open intervals can use a mark
Represents, where
。Similarly, half open interval can be used
or
express.
B-a is called the length of the interval.The length of an unbounded interval is
。R * itself can also be written
。In addition to the single point set, the interval is the only connected set in R. In the literature, "]" and "[" are often used to replace "(" and ")" respectively
writing
Similarly
Etc.[1]
Theorem of nested closed interval
Announce
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Closed interval sleeve
There are infinite closed intervals
, meet the following two conditions:
(1)[an+1,bn+1]⊆[an,bn](that is, the last closed interval is within the previous closed interval);
(2)
(that is, with the increase of n, the length of the closed interval becomes shorter and shorter),
Then the set formed by this infinite number of closed intervals
Is called a closedInterval sleeve, referred to as interval set.
theorem
if
Is a closed interval set, then there is a unique real number
, and
。
inference
If ξ is a closed interval set {[an,bn]}For any ε>0, there is always a natural number N; when n>N, there is [an,bn]⊂U(ξ,ε)。
That is, if ξ is a closed interval set {[an,bn]}Then, in the ε neighborhood of ξ, there is always an interval set {[an,bn]}Countless intervals of.
