## Characteristic functions and products of bounded derivatives.(English)Zbl 0833.26008

The author shows which characteristic functions can be expressed as the product of two or more bounded derivatives. The main result of the paper asserts: Let $$S \subset R$$ $$(R$$ – the real line) be ambiguous (i.e., both an $$F_\sigma$$ and a $$G_\delta$$ set) and $$T = R - S$$ be nonporous. Then there exist derivatives $$f,g$$: $$R \to R$$ such that (i) $$f\cdot g = 0$$ on $$T$$; (ii) $$f = g$$ and $$|f |= 1$$ on $$S$$; (iii) $$|f |< 2$$, $$|g |< 2$$.

### MSC:

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

### References:

 [1] Andrew M. Bruckner, Differentiation of real functions, Lecture Notes in Mathematics, vol. 659, Springer, Berlin, 1978. · Zbl 0382.26002 [2] Casper Goffman, C. J. Neugebauer, and T. Nishiura, Density topology and approximate continuity, Duke Math. J. 28 (1961), 497 – 505. · Zbl 0101.15502 [3] Casimir Kuratowski, Topologie. Vol. I, Monografie Matematyczne, Tom 20, Państwowe Wydawnictwo Naukowe, Warsaw, 1958 (French). 4ème éd. · Zbl 0078.14603 [4] Jan Mařík, Characteristic functions and products of derivatives, Real Anal. Exchange 16 (1990/91), no. 1, 245 – 254. · Zbl 0732.26007 [5] G. Petruska and M. Laczkovich, Baire 1 functions, approximately continuous functions and derivatives, Acta Math. Acad. Sci. Hungar. 25 (1974), 189 – 212. · Zbl 0279.26003 [6] Z. Zahorski, Sur la première dérivée, Trans. Amer. Math. Soc. 69 (1950), 1 – 54 (French). · Zbl 0038.20602
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.