Definition:Darboux Function

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.