Rethinking rigor in calculus: the role of the mean value theorem

An alternative to the Mean Value Theorem called the Increasing Function Theorem (IFT) is proposed. IFT states that if f' is greater than or equal to 0 on an interval, then f is increasing on that interval. IFT is claimed to be easier to state, more intuitive and more effective. The IFT shows how the theoretical structure of calculus can be made more thoughtful and sensible.