Intermediate Value Theorem/Corollary

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Theorem

Let $I$ be a real interval.

Let $f: I \to \R$ be a real function which is continuous on $\left({a \,.\,.\, b}\right)$.

Let $a, b \in I$ such that $\left({a \,.\,.\, b}\right)$ is an open interval.

Let $0 \in \R$ lie between $f \left({a}\right)$ and $f \left({b}\right)$.

That is, either:

$f \left({a}\right) < 0 < f \left({b}\right)$

or:

$f \left({b}\right) < 0 < f \left({a}\right)$


Then $f$ has a root in $\left({a \,.\,.\, b}\right)$.


Proof

Follows directly from the Intermediate Value Theorem and from the definition of root.

$\blacksquare$


Sources