This fact, which eventually belongs to Lebesgue, is usually proved with some measure theory (and we prove that the function is differentiable a.e.). Is there a significantly different approach? Let me explain how it could look like. Say, if the function is convex, we may touch its graph by a Euclidean disc (lying in the épigraphe), and in the point of touch there exists a derivative.

Maybe, it allows to prove something about the set of points where there is no derivative, not only that it has Lebesgue measure $0$. Or is this impossible and for any set $A$ of Lebesgue measure $0$ there exists a monotone function $f$ not differentiable at any point $a\in A$?

UPD: according to a comment by Bill Johnson, this statement is true, even for a Lipschitz function.

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