# How to prove that a monotone function is differentiable at some point?

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.

• There is a big literature on universal differentiability sets on spaces of dimension larger than one that have Lebesgue measure zero. On the line everything is known (a measurable subset of the line contains a point of differentiability of every Lipschitz function iff it has positive measure). The most recent paper on this topic with which I am familiar is Doré, Michael; Maleva, Olga A compact universal differentiability set with Hausdorff dimension one. Israel J. Math. 191 (2012), no. 2, 889–900. Dec 2, 2015 at 19:26
• A direct proof of Lebesgue's theorem, as well as the example requested in the last sentence, is given on pages 6-9 of the classic text Functional Analysis, by Riesz-Nagy, well before the discussion of integration or measure theory. Dec 2, 2015 at 20:19
• @roysmith thank you for the reference. Actually, I see two significantly different approaches: using Rising Sun Lemma (as in Riesz-Nagy) and using some covering theorem like Vitali theorem. In a board sense, they both use greedy choice of intervals, but somehow differently. And of course both they proof that function is differentiable in some point by proving that a.e. point works. While I wonder whether there is another way to find such a point. Well, I still have not seen Botsko's note mentioned in the answer by Igor Rivin. Dec 2, 2015 at 20:34
• Note that a.e. differentiability of monotone and of Lipschitz functions are strictly related. Indeed, if you rotate the graph of an increasing function 45 degrees you get a graph of a 1-Lipschitz function, and conversely. Dec 2, 2015 at 22:41
• @Pietro this is nice observation, but, say, points of a jump for $f$ produce points of locally linear function for the rotated function. Dec 2, 2015 at 22:54