In this MO question, Terrence Tao inquires about the everywhere differentiable inverse function theorem. This answer claims the theorem may be deduced from fairly intricate topological results of Cernavskii.
I've looked around and found the following two theorems which are claimed to generalize Cernavskii's results. I would like to understand how they're related, and how they imply the inverse function theorem.
The following is specifically introduced as a generalization of Cernavskii's results.
Theorem 1. [Thm 3] Let $(M,d)$ be a closed $n$-dimensional topological manifold with metric $d$ inducing the topology. There is an $\varepsilon >0$ such that if $Y$ is a closed topological manifold and $M\overset{f}{\to} Y$ is a proper surjection with finite fibers but not a homeomorphism, then there's at least one $y\in Y$ such that $\operatorname{diam}f^{-1}(y)\geq \varepsilon$ (the diameter is defined via the metric $d$ on $M$).
The second theorem lives in the setting of branched covering maps, and requires a preliminary definition to state.
Definition. A subset $A\subset X$ separates $X$ locally at $x\in X$ if there's a neighborhood $U\ni x$ such that for each neighborhood $V\ni x$ in $U$, the set $V\setminus A$ is not connected.
Theorem 2. Let $U\overset{f}{\to} V$ be an open map with discrete fibers between opens in $\mathbb R ^n$. Then the set on which $f$ is not a local homeomorphism has empty interior, and does not locally separate $U$ at any point.
Question 1. (How) Does each of the above theorems imply the inverse function theorem?
Question 2. What's the relation between the theorems? Does either imply the other?
