Intermediate Value Theorem/Corollary
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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