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The Intermediate Value Theorem

  • Titu Andreescu
  • Cristinel Mortici
  • Marian Tetiva
Chapter

Abstract

Let \(I \subseteq \mathbb{R}\) be an interval. We say that a function \(f: I \rightarrow \mathbb{R}\) has the intermediate value property (or IVP, for short) if it takes all intermediate values between any two of its values. More precisely, for every a, b ∈ I and for any λ between f(a) and f(b), we can find c between a and b such that f(c) = λ. A direct consequence of this definition is that f has IVP if and only if it transforms any interval into an interval. Equivalently, a function with IVP which takes values of opposite signs must vanish at some point.

Keywords

Direct Consequence Positive Integer Continuous Function Real Number Opposite Sign 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer Science+Business Media LLC 2017

Authors and Affiliations

  • Titu Andreescu
    • 1
  • Cristinel Mortici
    • 2
  • Marian Tetiva
    • 3
  1. 1.University of Texas at Dallas Natural Sciences and MathematicsRichardsonUSA
  2. 2.Valahia University of TargovisteTargovisteRomania
  3. 3.Gheorghe Rosca Codreanu National CollegeBarladRomania

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