The Intermediate Value Theorem

  • Titu Andreescu
  • Cristinel Mortici
  • Marian Tetiva


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.


Direct Consequence Positive Integer Continuous Function Real Number Opposite Sign 

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

Personalised recommendations