Intermediate value theorem

Statement

Full version

Suppose $f$ is a continuous function and a closed interval $[a,b]$ is contained in the domain of $f$ (in particular, the restriction of $f$ to the interval $[a,b]$ is continuous). Then, for any $t$ between the values $f(a)$ and $f(b)$ (see note below), there exists $c\in [a,b]$ such that $f(c)=t$.

Note: When we say $t$ is between $f(a)$ and $f(b)$, we mean $t\in [f(a),f(b)]$ if $f(a)\leq f(b)$ and we mean that $t\in [f(b),f(a)]$ if $f(b)\leq f(a)$.

Short version

Any continuous function on an interval satisfies the intermediate value property.