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.
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 $f$.