Location of Roots Theorem

Location of roots theorem is one of the most intutively obvious properties of continuos functions, as it states that if a continuos function attains positive and negative values, it must have a root.

Statement

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

Let $f$ be continuos on $[a,b]$

Let $f(a)<0$ and $f(b)>0$

Then $\exists c\in (a,b)$ such that $f(c)=0$

Proof

Let $A=\{x|x\in [a,b],\; f(x)<0\}$

As $a\in A$ , $A$ is non-empty. Also, as $A\subset [a,b]$ , $A$ is bounded

Thus $A$ has a Least upper bound, $\sup A=u\in A$ ...(1)

If $f(u)<0$ :

As $f$ is continous at $u$ , $\exists\delta>0$ such that $x\in V_{\delta}(u)\implies f(x)<0$ , which contradicts (1)

Also if $f(u)>0$ :

$f$ is continuos $\implies$ $\exists\delta>0$ such that $x\in V_{\delta}(u)\implies f(x)>0$ , which, by Gap lemma, again contradicts (1)

Hence, $f(u)=0$

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Location of Roots Theorem

Statement

Proof

See Also