Location of Roots Theorem

The location of roots theorem is one of the most intutively obvious properties of continuous functions, as it states that if a continuous function attains positive and negative values, it must have a root (i.e. it must pass through 0).

Statement

Let $f:[a,b]\rightarrow\mathbb{R}$ be a continuous function such that $f(a)<0$ and $f(b)>0$. Then there is some $c\in (a,b)$ 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, $\sup A = u \in A.$

If $f(u)<0$:

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

Hence, $f(u)=0$.