Some properties of multipliers of summable derivatives. (English) Zbl 0585.26005

Mařík, Jan

Real Anal. Exch. 9, 251-257 (1984).

If F is an arbitrary class of functions transforming the unit interval into real numbers, then M(F) denotes the system of all real functions f defined on the unit interval such that \(fg\in F\) for each \(g\in F.\) The functions in M(F) are called multipliers of F. The problem of characterizing of the system M (summable derivatives) has been solved by the author in Real Anal. Exch. 8, 486-493 (1983; Zbl 0554.26004). The main result of this paper states that the set of points of discontinuity of a function in M(SD), where SD is the class of all summable \((=Lebesgue\) integrable) derivatives, is countable and nowhere dense and that there exists a continuous function belonging to M(SD) which is nowhere differentiable.

Reviewer: J.Niewiarowski

MSC:

26A24	Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A27	Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
26A15	Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable

Keywords:

nowhere differentiable function; multipliers; summable derivatives; points of discontinuity

Citations:

Zbl 0554.26004