The equivalence of universal and ordinary first-return differentiation (Q1852349)

The author deals with functions \(F: \mathbb{R}\to\mathbb{R}\). Let \(S\) be a countable dense set of reals, which we call the ``support set''. Let \(\sigma: S\to\mathbb{Z}^+\) be an injection, or an ordering on \(S\), which is referred to as a ``trajectory''. For each \(s\in S\), \(\sigma(s)\) is called the rank of \(s\) (rank\((s)\)). The ``path system'' \(P\) denotes the relation on \(S\times\mathbb{R}\) defined by \((s,x)\in P\) if and only if \(s\neq x\) and no element \(r\in S\) between \(s\) and \(x\) has \(\text{rank}(r)< \text{rank}(s)\). For each real number \(x\), let path\((x)\) denote the set \(\{s\in S: (s,x)\in P\}\). The limiting process as \(y\to x\), \(y\in \text{path}(x)\), is called the ``\(\sigma\)-first-return limit''. The ``\(\sigma\)-first-return derivative'' of \(F\) at \(x\) means the \(\sigma\)-first-return limit of \((F(y)- F(x))/(y- x)\). The function \(F\) is said to be first-return differentiable (universally first-return differentiable) to a finite function \(f\) if there exist some support set \(S\) and (given any support set \(S\) there exists) some trajectory \(\sigma: S\to\mathbb{Z}^+\) such that at each \(x\), the \(\sigma\)-first-return derivative of \(F\) is \(f(x)\). The author proves: If \(F: \mathbb{R}\to\mathbb{R}\) is first-return differentiable to a finite function \(f\), then \(F\) is also universally first-return differentiable to \(f\).