Intermediate Value Theorem

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


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


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


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


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 $f(c)=k.$


See Also