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# 14.1 The Intermediate Value Property

14.1   Assumption (Intermediate value property 1.) Let be real numbers with , and let be a continuous function from to R such that and . Then there is some number such that

The intermediate value theorem was first proved in 1817 by Bernard Bolzano (1781-1848). However Bolzano published his proof in a rather obscure Bohemian journal, and his work did not become well known until much later. Before the nineteenth century the theorem was often assumed implicitly, i.e. it was used without stating that it was an assumption.

14.2   Definition ( is between and .) Let , and be real numbers with . We say that is between and if either or .

14.3   Corollary (Intermediate value property 2.) Let be a continuous function from some interval to R, such that and have opposite signs. Then there is some number between and such that .

Proof:     If the result follows from assumption 14.1. Suppose that . Let for all . then is a continuous function on and . It follows that there is a number such that , and then

14.4   Corollary (Intermediate value property 3.) Let be real numbers with , and let be a continuous function such that . Let be any number between and . Then there is a number such that .

14.5   Exercise. A Prove Corollary 14.4. You may assume that .

Next: 14.2 Applications Up: 14. The Inverse Function Previous: 14. The Inverse Function   Index
Ray Mayer 2007-09-07