Intermediate value theorem

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.