Converse Of Intermediate Value Theorem
A real-valued function \(f\) is said to have the intermediate value property if for every \( [a,b] \) in the domain of \(f\), and for every \[x \in \Big[\min\big(f(a), f(b)\big), \max\big(f(a), f(b)\big)\Big],\] there exists some \(c\in [a,b]\) such that \(f(c) = x\).
The intermediate value theorem states that if \(f\) is continuous, then \(f\) has the intermediate value property. Is the converse of this theorem true? That is, if a function has the intermediate value property, must it be continuous on its domain?