This is a classic theorem of Lebesgue's. I believe there is a proof in Royden's Real Analysis.
Can you help me proof the theorem: i f f is a monotonic function defined on an interval I, then f is differentiable almost everywhere on I, i.e. the set of numbers x in I such that f is not differentiable in x has Lebesgue measure zero.
This is a classic theorem of Lebesgue's. I believe there is a proof in Royden's Real Analysis.