Intermediate Value Theorem/Corollary

Theorem

Let $I$ be a real interval.

Let $a, b \in I$ such that $\openint a b$ is an open interval.

Let $f: I \to \R$ be a real function which is continuous on $\openint a b$.

Let $0 \in \R$ lie between $\map f a$ and $\map f b$.

That is, either:

$\map f a < 0 < \map f b$

or:

$\map f b < 0 < \map f a$


Then $f$ has a root in $\openint a b$.


Proof

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

$\blacksquare$


Sources

  • 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Path-Connectedness
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