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The Complexity Of Nowhere Differentiable Continuous Functions

Published online by Cambridge University Press:  20 November 2018

T. I. Ramsamujh
T. I. Ramsamujh*
Affiliation:
Florida International University, Miami, Florida
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It was not always clear that there could exist a continuous function which was differentiable at no point. (Such functions are now known as nowhere differentiable continuous functions. By “differentiable” we mean having a finite derivative.) In fact in 1806 M. Ampere [2] even tried to show that no such function could exist but his reasonings were later discovered to be fallacious. Of the early attempts at constructing a nowhere differentiable continuous function mention must be made of B. Bolzano. In a manuscript dated around 1830, (see [21]) he constructed a continuous function on an interval and showed that it was not differentiable on a dense set of points. (It was later shown by K. Rychlik [21] that this function was in fact nowhere differentiable.)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 41 , Issue 1 , 01 February 1989 , pp. 83 - 105
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Ajtai, M. and Kechris, A., The set of continuous functions with everywhere convergent Fourier series, Trans. Amer. Math. Soc. 302 (1987), 207–221.Google Scholar
2. Ampère, M., Recherches sur quelques points de la théorie des fonctions dérivées etc., Ecole Polytechnique, Band 6, Cahier 13 (1806), 148–181.Google Scholar
3. Banach, S., Über die Baire'sche Kategorie gewisser Funktionenmengen, Studia Math. 3 (1931), 174–179.Google Scholar
4. Besicovitch, A., Diskussion der steitgen Funktionen im Zusamenhang mit der Frage über ihre Dijferentiierbarkeit, Isvestiya Akad. Nauk SSSR 19 (1925), 527–540.Google Scholar
5. Cellerier, C., Note sur les principes fondamentaux de l'analyse, Bull, des Sci. Math., ser. 2, 4 (1890), 142–160.Google Scholar
6. Du Bois-Reymond, P., Versuch einer Klassification der willKurlurchen Funktionen reeler Argumente, J. fur Math. 79 (1874), 21–37.Google Scholar
7. Gerver, J., The differentiability of the Riemann function at certain rational multiples of pi, Amer. J. of Math. 92 (1970), 35–55.Google Scholar
8. Gerver, J., More on the differentiability of the Riemann function, Amer. J. of Math. 93 (1971) 33–41.Google Scholar
9. Kechris, A., Sets of everywhere singular functions, Lecture Notes in Math. 1141 (Springer- Verlag, Berlin, 1985), 233–244.Google Scholar
10. Kechris, A. and Woodin, H., Ranks of differentiable functions, Mathematika 33 (1986), 252–278.Google Scholar
11. Knopp, K., Fin einfaches Verfahren zur Bildung stetiger nirgends differenzier barer Funktionen, Math Zeit. 2 (1918), 1–26.Google Scholar
12. Kowalewski, G., Ueber Bolzanos Nicht-dijferenzierbare stetige Funktion, Acta Math. 44 (1923), 315–319.Google Scholar
13. Malý, J., Where the continuous functions without unilateral derivatives are typical, Trans. Amer. Math. Soc. 283 (1984), 169–175.Google Scholar
14. Mauldin, R., The set of continuous nowhere differ entiable functions, Pacific J. of Math. 83 (1979), 199–205.Google Scholar
15. Mauldin, R., The set of continuous nowhere differ entiable function: A correction, Pacific J. of Math. 121 (1986), 119–120.Google Scholar
16. Mazurkiewicz, S., Sur les fonctions non derivables, Studia Math. 3 (1931), 92–94.Google Scholar
17. Morse, A., A continuous function with no unilateral derivatives, Trans. Amer. Math. Soc. 44 (1938), 496–507.Google Scholar
18. Moschovakis, Y.N., Descriptive set theory (North Holland Publ., New York, 1980).Google Scholar
19. Pepper, E.D., On continuous functions without derivatives, Fund. Math. 12 (1928), 244—253.Google Scholar
20. Ramsamujh, T.I., Some topics in descriptive set theory and analysis, Ph.D. thesis, Calif. Inst. of Tech., Pasadena, California (1986).Google Scholar
21. Rychlik, K., Ueber eine Funktion aus Bolzanos hand-schriftlichen Nachlasse, Ceska Spolenost nauk, Prague Trida matematicko pritodoved Vestik (1921-22), 1–20.Google Scholar
22. Saks, S., On the functions of Besicovitch in the space of continuous functions, Fund. Math. 19 (1932), 211–219.Google Scholar
23. Singh, A.N., On Bolzano's non-differ entiable functions, Bull, de l'Acad. Polonaise des Sci. (1928), 191–220.Google Scholar
24. Singh, A.N., The theory and construction of non-differentiable functions, Lucknow University Studies, 1 (1935).Google Scholar
25. Smith, A., The differentiability of Riemann s function, Proc. Amer. Math. Soc. 34 (1972), 463–468.Google Scholar
26. Takagi, T., A simple example of a continuous function with no derivative, Proc. Physico-Math. Soc. of Japan, Ser. II, No. 1 (1903), 176–177.Google Scholar