# Monotone functions are differentiable a.e. and Hilbert's Fifth Problem: what's the connection?

In Andrew Gleason's interview for More Mathematical People, there is the following exchange concerning Gleason's work on Hilbert's fifth problem on whether every locally Euclidean topological group is a Lie group (page 92).

MP: Is there some "human" story you can tell us about the breakthrough when it came?

Gleason: Yes, there's a really remarkable story about that. Sometime -- I can't tell you the exact date but let's say around 1949 -- I was doing other things too, and one of the things that I found very interesting and very curious and which I really felt I should try to understand better was a very famous theorem to the effect that a monotonic function is almost everywhere differentiable. It's a rather remarkable and very difficult theorem -- it's not easy to prove. A very very hard theorem of analysis and a really surprising theorem. Well, at the time I was sort of speculating about this theorem, but it wasn't for at least two years that I suddenly realized that that would solve the problem I was dealing with! Knowing that, in connection with some other stuff I had been working on, really put the whole thing together. It was a realization that although this theorem had been on my mind for maybe two years, I had never recognized that it was crucial to the arguments that I was trying to work through in the Hilbert problem. I hadn't realized it. Then suddenly it just came to me.

MP: It just came to you?

Gleason: That's right. It just came to me that I could use this technique, this theorem, in connection with these curves in Hilbert space that I was dealing with -- and get the answer! ...

I've never studied Hilbert's Fifth Problem or its solutions, but I've always been curious what Gleason meant by this connection. Can anyone shed some light on this?

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Just a curious question in passing, whether the flip-side: Everywhere differentiable, but nowhere monotone functions, are also of interest? See e.g., jstor.org/stable/2318996 – Suvrit Nov 2 '10 at 9:46
I don't see how it is relevant here, but such wiggly things are indeed curious and interesting. – Todd Trimble Nov 2 '10 at 11:22

@Todd: "...it's more ideas from the proof of this theorem of analysis than the statement itself that is relevant". Yes. Recall from my article that, in attempting to define a $1$-parameter group $f$ with $f(1)$ a given group element near the identity, from the existence of unique square roots near $e$ it is easy to see how to define $f$ at all dyadic rationals $m/2^n$. More difficult is to see how to prove that $f$ so defined is continuous---and so extends to a $1$-parameter group. This is where I think Gleason is saying that he used ideas from the proof of the monotone function theorem. – Dick Palais Nov 2 '10 at 16:32