Continuous functions have the intermediate value property

Corollary (Intermediate Value Theorem): Let $I\subseteq\mathbb{R}$ be an interval and let $f:I\to\mathbb{R}$ be continuous . If $a,b\in I$ and $y$ lies between $f(a)$ and $f(b)$, then there exists $c$ between $a$ and $b$ such that $f(c)=y.$

Connection to parent theorem: This is the intermediate value theorem , which follows from connectedness of intervals and the fact that continuous images of connected sets are connected subsets of $\mathbb{R}$, hence intervals.