Intermediate Value Theorem

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The Intermediate Value Theorem is one of the very interesting properties of continous functions.

Statement

Take a function $f$ and interval $[a,b]$ such that the following hold:

$f:[a,b]\rightarrow\mathbb{R},$

$f$ is continuous on $[a,b],$

$f(a)<k<f(b).$

Then, $\exists c\in (a,b)$ such that $f(c)=k.$

Proof

Consider $g:[a,b]\rightarrow\mathbb{R}$ such that $g(x)=f(x)-k.$

Note that $g(a)<0$ and $g(b)>0$

By the Location of roots theorem, $\exists c\in (a,b)$ such that $g(c)=0$ or $f(c)=k.$

QED

See Also

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