Where the continuous functions without unilateral derivatives are typical

Malý, Jan

doi:10.1090/S0002-9947-1984-0735414-9

Where the continuous functions without unilateral derivatives are typical
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by Jan Malý PDF

Trans. Amer. Math. Soc. 283 (1984), 169-175 Request permission

Abstract:

An alternative proof of the existence of a Besicovitch function (i.e. a continuous function which has nowhere a unilateral derivative) is presented. The method consists in showing the residuality of Besicovitch functions in special subspaces of the Banach space of all continuous functions on $[0,1]$ and yields Besicovitch functions with additional properties of Morse or Hölder type. A way how to obtain functions with a similar behavior on normed linear spaces is briefly mentioned.

References

Über die Baire’ sche Kategorie gewisser Funktionenmengen

3

Discussion der stetigen Funktionen im Zusammenhang mit der Frage über ihre Differenzierbarkeit

19

K. M. Garg, On asymmetrical derivates of non-differentiable functions, Canadian J. Math. 20 (1968), 135–143. MR 220878, DOI 10.4153/CJM-1968-015-1
R. L. Jeffery, The Theory of Functions of a Real Variable, Mathematical Expositions, No. 6, University of Toronto Press, Toronto, Ont., 1951. MR 0043162

Sur les fonctions non dérivables

3

Anthony P. Morse, A continuous function with no unilateral derivatives, Trans. Amer. Math. Soc. 44 (1938), no. 3, 496–507. MR 1501978, DOI 10.1090/S0002-9947-1938-1501978-2

On continuous functions without a derivative

12

On the functions of Besicovitch in the space of continuous functions

19

A. N. Singh, On functions without one-sided derivatives, Proc. Benares Math. Soc. (N.S.) 3 (1941), 55–69. MR 9623
A. N. Singh, On functions without one-sided derivatives. II, Proc. Benares Math. Soc. (N.S.) 4 (1943), 95–108. MR 9979
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776