Definition:Intermediate Value Property

Definition

Let $f: \R \to \R$ be a real function.

Then $f$ has the intermediate value property if, given any $\, b, d \in \R$ such that $a < b$ and $d$ is between $f \left({a}\right)$ and $f \left({b}\right)$, there exists at least one $c \in \R$ such that $a \le c \le b$ and $f \left({c}\right) = d$.

Thus, for every intermediate value between $f \left({a}\right)$ and $f \left({b}\right)$, that value is the image of some intermediate value between $a$ and $b$.

This property is frequently seen abbreviated I.V.P.

Sources

• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Definition $1.4.1$