Intermediate Value Theorem

Intermediate Value Theorem is a basic theorem in calculus. Usually, there is nothing more to say about the theorem. But with the rigorous bounds obtained in numerical algorithms that we discussed before, much can be done through the basic theorem. In mathematical research, existence is often very hard to establish. Part of the reason is most existences afford no constructive description. Here we are left with indirect methods like the Intermediate Value Theorem. Traditionally, numerical algorithms offered insights but were not applicable in rigorous proof. With rigorous bounds proven, numerical algorithms can be utilized to provide rigorous estimates of simulated values. Together with the likes of Intermediate Value Theorem, existence of objects can be established rigorously.

Providing rigorous bounds for numerical algorithms is a game changer.