Intermediate Value Theorem

Intermediate Value Theorem: Let $I\subseteq\mathbb{R}$ be an interval and let $f:I\to\mathbb{R}$ be continuous . If $a,b\in I$ with $a<b$ and $y$ is any real number between $f(a)$ and $f(b)$, then there exists $c\in[a,b]$ such that $f(c)=y.$

This theorem formalizes the idea that continuous functions on intervals cannot “jump over” values, and it is the basis for existence of roots, fixed points on intervals, and many approximation results.

Proof sketch (optional): Consider $S=\{x\in[a,b]: f(x)\le y\}$ (assuming $f(a)\le y\le f(b)$). Let $c=\sup S$. Continuity shows $f(c)=y$ by ruling out $f(c)<y$ and $f(c)>y$ via neighborhood arguments.