Fundamental proof of calculus and mean value theorem

Question

Hi I have been trying to learn the proof for the fundamental theorem of calculus and I found out that I need to learn the mean value theorem. However this theorem is quite strange as some proofs say that x needs to be differentiable only on the open interval $(a,b)$, and other proofs such as the one on khan academy state that x is differentiable on the closed interval. There seems to be lots of contradiction online, so is there a intuitive way to prove the fundamental theorem of calculus?