Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that if $f$ is a continuous function whose domain contains the interval $[a, b]$ , then it takes on any given value between $f(a)$ and $f(b)$ at some point within the interval.

This has two important corollaries:

If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem).^[1] ^[2]
The image of a continuous function over an interval is itself an interval.

^ Weisstein, Eric W. "Bolzano's Theorem". MathWorld.
^ Cates, Dennis M. (2019). Cauchy's Calcul Infinitésimal. p. 249. doi:10.1007/978-3-030-11036-9. ISBN 978-3-030-11035-2. S2CID 132587955.

[1] Weisstein, Eric W. "Bolzano's Theorem". MathWorld.

[2] Cates, Dennis M. (2019). Cauchy's Calcul Infinitésimal. p. 249. doi:10.1007/978-3-030-11036-9. ISBN 978-3-030-11035-2. S2CID 132587955.

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