## 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
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## Copyright information

© Springer Science+Business Media LLC 2017