I am reading Maly's proof of Stepanov's theorem which asserts the following:
Theorem Let $\Omega \subset \mathbb{R}^n$ be open, and let $f: \Omega \rightarrow \mathbb{R}^m$ be a function. Then $f$ is differentiable almost everywhere on $$L(f) : = \{x\in \Omega : \text{Lip}f(x) < \infty\},$$
where $$\text{Lip}f(x): = \limsup_{y \to x} \frac{|f(y)- f(x)|}{|y-x|} .$$
(for example, see Theorem 3.4 in Lectures on Lipschitz Analysis).
Proof: We may assume that $m=1$. Let $\{B_1, B_2, \ldots\}$ be the countable collection of balls contained in $\Omega$ such that each $B_i$ has rational center and radius, and $f_{|{B_i}}$ is bounded. In particular this collection covers $L(f)$. Define $$u_i(x): = \inf \{u(x) : u \, \text{is $i$-Lipschitz with $u\geq f$ on $B_i$}\} $$ and $$v_i(x): = \sup \{v(x) : u \, \text{is $i$-Lipschitz with $v\leq f$ on $B_i$}\}. $$ [...]
In the proof, we say that "Is it clear that $f$ is differentiable at every point $a$, where, for some $i$, both $u_i$ and $v_i$ are differentiable with $v_i(a) = u_i(a)$" (see page 24).
I would like to verify this claim.
If the derivative of $u_i$ and $v_i$ at point $a$ are equal, it seems to be clear, but what about in case that they are not equal?
Any suggestion? Thanks in advance!
Let $\{e_j\}$ denote the standard basis vectors in ${\mathbb R}^n$.
Suppose that both $u_i$ and $v_i$ are differentiable at $a \in B_i$, with $v_i(a) = u_i(a)$.
Let $\psi=u_i-v_i$, so that $\psi \ge 0$ on $B_i$.
If $\frac{\partial \psi}{\partial x_j}(a) >0$, then $\psi(a-\epsilon e_j)<0$ for small enough $\epsilon>0$, a contradiction. Similarly, if $\frac{\partial \psi}{\partial x_j}(a) <0$, then considering $\psi(a+\epsilon e_j)$ for small $\epsilon>0$ yields a contradiction. Thus $$ 0=\nabla \psi=\nabla u_i-\nabla v_i \,.$$