Continuity implies the intermediate value property How can someone prove that continuity implies the intermediate value property?
P.S.: If $I$ is an interval, and $f:I\rightarrow\mathbb{R}$, we say that $f$ has the intermediate value property (IVP) iff whenever $a<b$ are points in $I$ and $f(a)\leq c\leq f(b)$, there is a $d$ between $a$ and $b$ such that $f(d)=c$.

I guess it depends on the logical framework you're using.
In smooth infinitesimal analysis (which uses intuitionistic logic rather than classical logic), every function is continuous, and the intermediate value theorem fails!
