## The equivalence of universal and ordinary first-return differentiation.(English)Zbl 1026.26004

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 $$f$$.

### MSC:

 26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems