TITLE: reverse mathematics strength of "Lipschitz functions are somewhere differentiable"
QUESTION [6 upvotes]: What is the reverse mathematics strength of

"For all Lipschitz functions $\; f : \mathbb{R} \to \mathbb{R} \;$, $\;$ there exists a real number $x$ such that $f$ is differentiable at $x$." ?

(defined using epsilon-delta, so not requiring that there exist a function witnessing the convergence)
Since Lipschitz functions are differentiable almost everywhere,

I would guess the answer is "is equivalent to WWKL$_0$ (over RCA$_0$)".

REPLY [4 votes]: Here it is, the newest version
http://dl.dropbox.com/u/370127/papers/randomnessanalysisjuly_2011.pdf
What jason says is right, even for nondecreasing functions. It is in our paper, Subection 5.1
Hence there is (I believe) a real $y$ computable from the function $f$ for which $f$ is differentiable at $x$.
Hence your theorem holds in every $\omega$-model of RCA0.
This implies that your theorem is true in RCA0 or RCA0 plus some first order principle.
Not sure what these fo principles do, I guess it's standard stuff?