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Proceedings of the American Mathematical Society

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Differentiability through change of variables


Authors: A. M. Bruckner and C. Goffman
Journal: Proc. Amer. Math. Soc. 61 (1976), 235-241
MSC: Primary 26A24
DOI: https://doi.org/10.1090/S0002-9939-1976-0432831-2
MathSciNet review: 0432831
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Abstract: A real function $ f$ on $ [0,\;1]$ can be transformed by a homeomorphism into a differentiable function with bounded derivative if and only if $ f$ is continuous and of bounded variation. This condition does not suffice for $ f$ to be transformed into a continuously differentiable function. The additional condition for this to hold is found and the theorem is proved.


References [Enhancements On Off] (What's this?)

  • [1] A. M. Bruckner and John. L. Leonard, On differentiable functions having an everywhere dense set of intervals of constancy, Canad. Math. Bull. 8 (1965), 73-76. MR 30 #4870. MR 0174670 (30:4870)
  • [2] S. Saks, Theory of the integral, Monografie Mat., Warsaw, 1933; rev. ed., English transl., Stechert, New York, 1937.
  • [3] Z. Zahorski, Sur la première dérivée, Trans. Amer. Math. Soc. 69 (1950), 1-54. MR 12, 247. MR 0037338 (12:247c)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0432831-2
Keywords: Homeomorphism, differentiable function, bounded variation, variable monotonicity
Article copyright: © Copyright 1976 American Mathematical Society

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