The Increasing Function Theorem (IFT), which connects positive derivatives and increasing functions, is absolutely foundational to a first course in calculus: for example, it justifies the use of the first derivative test for extreme values. The theorem feels intuitively obvious (positive slope means increasing, right?) but proving it is harder than one might expect. For this reason, many books “prove” the IFT using the Mean Value Theorem (MVT). But this can hardly be called rigorous if the MVT is left unproved? Even worse is to prove the MVT using the Extreme Value Theorem, but then leave that unproved. If we’re going to tell students to be suspicious of their intuition, we need to give them something better in its place.
One can simply say to students “the IFT is true, but its proof is harder than you might expect. It is not easy to relate the properties of a function across a whole interval to its properties at individual points in that interval.” Then there is no need to muddy the waters using unproved theorems to justify a claim that students probably found intuitive in the first place.
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