TITLE: How to prove that a monotone function is differentiable at some point?
QUESTION [12 upvotes]: 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.

REPLY [10 votes]: Yes, there are a couple of alternate proofs: Here is one:
An Elementary Proof of Lebesgue's Differentiation Theorem
Michael W. Botsko
The American Mathematical Monthly
Vol. 110, No. 9 (Nov., 2003), pp. 834
And this proof by Faure (from the Real Analysis exchange, 2003), which uses Riesz's 1932 idea: http://projecteuclid.org/download/pdf_1/euclid.rae/1149698560