I studying the extension problem for classes of function $C^0,C^1,D^1$ ($D^1=$class of derivable function). For $C^0$ function there is Tietze theorem and for $D^1$ function there is Jarnik theorem (rediscovered by G. Petruska and M.Laczkovich-https://math.wvu.edu/~kciesiel/prepF/129.DifferentiableExtensionThm/129.DifferentiableExtensionThm.pdf).
I pose attention on $C^1$ functions.
I know that: exists a function $f\in C^1(X)$ on a perfect set $X ⊂ \mathbb{R}$ with no has extension $F \in C^1(\mathbb{R})$.
I'm looking for a necessary and sufficient condition for existence of a extension of $f\in C^1(X)$ where $X\subseteq\mathbb{R}$ is perfect.
I have found a very general Whitney's theorem that i want adapte to monodimensional case. I ask: it's true that $f\in C^1(X)$ is extendible to $F\in C^1(\mathbb{R})$ iff $\forall \varepsilon>0$ $\exists\delta>0$ t.c. $\lvert f(x)-f(y)-f'(y)(x-y)\rvert<\varepsilon\lvert x-y\rvert$ $\forall x,y\in P$ ,$\lvert x-y\rvert<\delta$ ?
Is this correct? Can someone kindly show me a reference or an idea of the proof (aslike Jarnik?) of this property?
