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On Asymmetrical Derivates of Non-Differentiable Functions

Published online by Cambridge University Press:  20 November 2018

K. M. Garg
K. M. Garg*
Affiliation:
University of Alberta, Edmonton, Alberta
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Let ƒ(x) be a non-differentiable function, i.e. a realvalued continuous function denned on a linear interval which has nowhere a finite or infinite derivative. We shall say that ƒ(x) has symmetrical derivates at a point x if the four Dini derivates of ƒ(x) at x satisfy the relations

and otherwise we shall say that ƒ(x) has asymmetrical dérivâtes at x.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 135 - 143
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Besicovitch, A. S., Discussion der stetigen Funktionen im Zusammenhang mit der Frage ilber ihre Differenzierbarkeit, Bull. Acad. Sci. URSS, 19 (1925), 527–540.Google Scholar
2. Denjoy, A., Mémoire sur les nombres dérivés des fonctions continues, J. Math. Pures Appl. Ser. 7), 1 (1915), 105–240.Google Scholar
3. Garg, K. M., An analogue of Denjoy's theorem, Ganita, 12 (1961), 9–14.Google Scholar
4. Garg, K. M., On nowhere monotone functions II {Dérivâtes at sets of power c and at sets of positive measure), Rev. Math. Pures Appl., 7 (1962), 663–671.Google Scholar
5. Garg, K. M., On nowhere monotone functions III {Functions of first and second species), Rev. Math. Pures Appl., 8 (1963), 83–90.Google Scholar
6. Garg, K. M., Applications of Denjoy analogue III {Distribution of various typical level sets), Acta Math. Acad. Sci. Hungar., 22 (1963), 187–195.Google Scholar
7. Jarnik, V., Ûber die Differenzierbarkeit stetiger Funktionen, Fund. Math., 21 (1933), 48–58.Google Scholar
8. Jeffery, R. L., The theory of functions of a real variable (2nd éd., Toronto, 1953).Google Scholar
9. Morse, A. P., A continuous function with no unilateral derivatives, Trans. Amer. Math. Soc, 22 (1938), 496–507.Google Scholar
10. Morse, A. P., Dini derivatives of continuous functions, Proc. Amer. Math. Soc, 5 (1954), 126–130.Google Scholar
11. Pepper, E. D., On continuous functions without a derivative, Fund. Math., 12 (1928), 244–253.Google Scholar
12. Porter, M. B., Derivateless continuous functions, Bull. Amer. Math. Soc, 25 (1919), 176–180.Google Scholar
13. Saks, S., On the functions of Besicovitch in the space of continuous functions, Fund. Math., 19 (1932), 211–219.Google Scholar
14. Singh, A. N., Some new theorems on the dérivâtes of a function, Bull. Calcutta Math. Soc, 16 1925/26), 101–108.Google Scholar
15. Singh, A. N., On the dérivâtes of a function, Bull. Calcutta Math. Soc, 22 (1930), 1–8.Google Scholar
16. Singh, A. N., On the existence of a derivative, Bull. Calcutta Math. Soc, 23 (1931), 45–56.Google Scholar
17. Singh, A. N., The theory and construction of non-differentiable functions (Lucknow, 1935).Google Scholar
18. Singh, A. N., Analytical consideration of Besicovitch's function without one-sided derivatives, Proc. Benares Math. Soc. (N.S.), 3 (1941), 55–69.Google Scholar
19. Singh, A. N., On functions without one-sided derivatives II, Proc. Benares Math. Soc. (N.S.), 4 1942), 95–108.Google Scholar
20. Young, G. C., On infinite dérivâtes, Quart. J. Math. (Oxford), 47 (1916), 127–175.Google Scholar
21. Young, W. H., On the dérivâtes of non-differentiable functions, Messenger of Math., 38 1908), 65–69.Google Scholar