# Monotonic function differentiable almost everywhere

This result tells us that every monotone function (increasing or decreasing) defined on an open interval is differentiable almost everywhere on. Differentiability of Monotone Functions In this section we prove that a monotone function on an open interval almost everywhere on (a, b). up vote 10 down vote. Even any function of bounded variation is differentiable almost everywhere. share|cite|improve this answer. answered Jan 11 '11 at

Yes, there are a couple of alternate proofs: Here is one: An Elementary Proof of Lebesgue's Differentiation Theorem Michael W. Botsko The. then f' exists and is zero almost everywhere. Since each monotonic function is the sum of a monotonic continuous function and a jump function, the result for. Lebesgue established that a continuous monotone function is differentiable almost everywhere. To prove this, in technical parts we need: • A covering lemma .

We establish a sharp integrability condition on the partial derivatives of a weakly monotone Sobolev function to guarantee differentiability almost everywhere. Conversely, every such function defines a uniquely determines a borel measure on. (a, b]. . that D−F(x) ≤ D+F and thus F is differentiable almost everywhere.). tives of a weakly monotone Sobolev function to guarantee differentia- bility almost Ω C Rn, for some p > n, then u is differentiable almost everywhere in Ω. is almost everywhere differentiable, show that the (almost . assert for instance that monotone, Lipschitz, or bounded variation functions in one.

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