Properties of derivative-like functions. (English) Zbl 0942.26013

Bullen, P. S.; Sarkhel, D. N.

Real Anal. Exch. 22(1996-97), No. 2, 740-759 (1997).

The authors study the relations between the properties of generalized derivatives and the properties of ordinary derivatives, e.g., Darboux, Baire-1, Zahorski etc. Among others let us mention two theorems.
Let \(T(x)\) be a limiting system on \(I\) (i.e., the collection of nonempty sets having \(x\in I\) as a two-sided limit point and satisfying \(A\in T(x) \Rightarrow A\cap [x-\alpha ,x+\beta ]\in T(x)\) for all \(\alpha ,\beta >0\)). Let \[ T\overline {F}'(x) = \inf _{A\in T(x)} \sup \left \{\frac {F(v)-F(u)}{v-u} \^^M;\;u\leq x\leq v,\;u,v\in A,\;u<v\right \}. \] Similarly \(T\underline {F}'(x)\). The common value \(TF'(x)\) is called \(T\)-derivative. Let \(p(x,A)\) denote the porosity of \(A\) in \(x\).
1. If \(T\) satisfies ILC (interlocking condition) and every \(A\in T(x)\) has \(p(x,A)=0\) then \(F'(x)\) exists finitely for \(x\) in a dense open set in \(I\) provided \[ -\infty <T\underline {F}'(x)\leq T\overline {F}'(x)<+\infty . \]
2. If \(T\) satisfies ILC and is non-porous in generalized sense (defined in the paper) then every \(f\) verifying \(T\overline {F}'(x) \leq f(x) \leq T\underline {F}'(x)\) on \(I\) is semi-Baire 1 on \(I\) (i.e., every level set \(f^{-1}(t)\) contains a point of continuity of \(f\) relative to the closure of \(f^{-1}(t)\)).

Reviewer: Jaroslav Tišer (Praha)

MSC:

26A24

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

Keywords:

generalized derivatives; porosity; partitioning covers; Baire-1; interlocking; Zahorski; Darboux