A counterexample for a smoth version of Tietze extension theorem Is there any function $f:F\subset \mathbb{R}^2\rightarrow \mathbb{R}$ with $F$ closed such that $f|F$ is differentiable in every accumulation point but there is no differentiable extension to the entire plane?
I think that such function exists but I can't find any.
 A: $\newcommand{\R}{\mathbb{R}}$
Smooth Version of Tietze Extension theorem: Let $F$ be any closed subset of
$\R^n$ and $f : F \to \R$ be a smooth map. Then there exists a smooth map $g : \R^n  \to \R$ such
that $g|_F = f$. This can be proved very easily, by using smooth version of partition of
unity for subspaces of euclidean spaces. Similarly, smooth version of Urysohn’s lemma
is also true.
However, the strict version of Tietze’s extension theorem in which the original function $f$ as well its extension $g$ in the conclusion take values inside $[0, 1]$, is FALSE.
Consider
the identity map $f : [0, 1] \to [0, 1]$ which is obviously smooth. Suppose $g : R \to [0, 1]$ is
a $C^1$ -map such that $g(x) = x$ for $0 \leq x \leq 1$. Then $g'(0) = 1$ as computed by taking the
right-hand derivative. Therefore for some positive $r$,  $g'(x)$ is positive for all $−r < x < r$.
This means g is strictly monotonically increasing in $(-r, r)$. Since $g(0) = 0, g(x) < 0$ for
$−r < x < 0$ which is absurd, because $g(\R) \subset [0, 1]$.
A: I have partial, but positive results. 
There is a fundamental book “Differential and Integral Calculus” by Grigorii Fichtenholz. This is a famous book for our students and it has many translations (but except English).
I found in Appendix of the vol. I the following Theorem (I-II), with a long and complex proof. 
A simple curve without special points represented by an equation $x=\varphi(t)$, $y=\psi(t)$, where $t$ belongs to an interval, is called a smooth curve of the class $C^n$ ($n\ge 1$), if the functions $\varphi$ and $\psi$ belongs in the interval to the class $C^n$.
Theorem. If a function $f(x,y)$ belongs to the class $C^n$ ($n\ge 1$) in a bounded closed domain, with a boundary consisting of one or several (non-intersecting) piecewise smooth simple curves (from the class $C^n$ too) then the function $f$ can be extended onto the entire plane $\Bbb R^2$ with the preservation of the class.
Another extension construction is suggested by $\S$ 3 of Ch. 15, vol. III, in which, in particular, is solved a problem when an expression $P dx + Q dy$  is an exact differential of a function $f$ (it is, under some assumptions (in particular, the existence and continuity of derivatives $\frac{\partial P}{\partial y}$ and $\frac{\partial Q}{\partial x}$), the equality $\frac {\partial P}{\partial y}=\frac {\partial Q}{\partial x}$ ). So, assume that a derivative $g=\frac {\partial^2 f}{\partial x\partial y}$ is continuous. Then, by Tietze extension theorem it can be extended onto the entire plane, onto a continuous function $\hat g$. Now I expect that by some simple integrals of the function $\hat g$ we can “recover” an extension of the function $f$.
