Bolzano's Root Point Theorem

analysis

Definition

Bolzano's Root Point Theorem

Let $f:[a,b]\to \mathbb{R}$ be a continuous function on a closed interval $[a,b]$. If $f(a)$ and $f(b)$ have opposite signs (i.e.,$f(a)\cdot f(b)<0$), then there exists at least one value $c\in (a,b)$ such that $f(c)=0$.

Significance

Bolzano’s theorem is a special case of the Intermediate Value Theorem. It provides a theoretical basis for numerical root-finding algorithms, such as the bisection method, which iteratively narrows down the interval containing the root.