On extendable derivations (Q1029989)

Given a function \(f:{\mathbb R}\to{\mathbb R},\) the author investigates some of its possible aspects, as being {\parindent=6mm \begin{itemize}\item[i.] additive (\(f\in \text{Add}\)), i.e. \(f(x+y)=f(x)+f(y)\) for all \(x,y\in{\mathbb R};\) \item[ii.] a derivation (\(f\in \text{Der}\)), i.e. \(f\in \text{Add}\) and \(f(xy)=xf(y)+yf(x)\) for all \(x,y\in{\mathbb R};\) \item[iii.] almost continuous in the sense of Stallings (\(f\in \text{ACS}\)), i.e. every open neighbourhood of \(f\) in \({\mathbb R}^2\) contains also a continuous function \(g:{\mathbb R}\to{\mathbb R};\) \item[iv.] extendable (\(f\in \text{Ext}\)), if there is a connectivity function \(F:{\mathbb R}\times [0,1]\to{\mathbb R}\) such that \(F(x,0)=f(x)\) when \(x\in{\mathbb R}.\) \end{itemize}} In the first part of the note, in Theorem 4, it is proved that there exists an almost continuous derivation that is not extendable. In the second part, the author considers algebraic properties of the classes \(\text{Der}\cap \text{Ext}\) and \(\text{Add}\cap \text{Ext}.\) In particular, he proves that each function \(f\in \text{Der}\) can be represented as: the sum of two functions in \(\text{Ext}\cap \text{Der};\) the pointwise limit of a sequence of functions in \(\text{Ext}\cap \text{Der};\) the limit of a transfinite sequence of functions from the class \(\text{Ext}\cap \text{Der}.\) Similar results hold for the class \(\text{Add}.\)