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Concept
Intermediate value theorem
Summary
In mathematical analysis, the intermediate value theorem states that if is a continuous function whose domain contains the interval , then it takes on any given value between and at some point within the interval.
This has two important corollaries:
If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem).
The of a continuous function over an interval is itself an interval.
This captures an intuitive property of continuous functions over the real numbers: given continuous on with the known values and , then the graph of must pass through the horizontal line while moves from to . It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the paper.
The intermediate value theorem states the following:
Consider an interval of real numbers and a continuous function . Then
Version I. if is a number between and , that is, then there is a such that .
Version II. the is also an interval (closed), and it contains ,
Remark: Version II states that the set of function values has no gap. For any two function values with , even if they are outside the interval between and , all points in the interval are also function values,
A subset of the real numbers with no internal gap is an interval. Version I is naturally contained in Version II.
The theorem depends on, and is equivalent to, the completeness of the real numbers. The intermediate value theorem does not apply to the rational numbers Q because gaps exist between rational numbers; irrational numbers fill those gaps. For example, the function for satisfies and . However, there is no rational number such that , because is an irrational number.
The theorem may be proven as a consequence of the completeness property of the real numbers as follows:
We shall prove the first case, . The second case is similar.
Let be the set of all such that . Then is non-empty since is an element of . Since is non-empty and bounded above by , by completeness, the supremum exists.
Official source
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