Rolle's theorem requirements

Question

To use Rolle's Theorem does a student have to must continuity then differentiability before any other parts? My son answered a question that Rolle's Theorem does not apply because f(a) does not equal f(b) which is another part of theorem. The answer the teacher wanted was that it was not continuous and that he had to determine continuity first. If all three criteria need to be met, what does it matter which order they are tested in?

Answer 1

First of all, differentiability at a point implies continuity at that point, but Rolle's theorem requires

continuity on [a,b], differentiability on (a,b) and f(a) = f(b).

The order in which these requirements is stated is NOT critical, but all 3 hypotheses ARE critical.