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Differentiability through change of variables. (English) Zbl 0358.26006

The authors establish the following result: 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. They remark that this condition does not suffice, however, for \(f\) to be transformable into a function with a continuous derivative. Then they prove that under the additional condition that the set \(M\) of points of varying monotonicity of \(f\) (a point \(x\in [0,1]\) is said to be a point of varying monotonicity if there is no neighborhood of \(x\) on which \(f\) is either strictly monotonic or strictly constant) has the property that \(\mu (f(M))= 0\) with \(\mu\) the Lebesgue measure, \(f\) can be transformed by a homeomorphism into a continuously differentiable function.
Reviewer: A. Precupanu

MSC:

26A45 Functions of bounded variation, generalizations
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
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References:

[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. · Zbl 0144.05103 · doi:10.4153/CMB-1965-009-1
[2] S. Saks, Theory of the integral, Monografie Mat., Warsaw, 1933; rev. ed., English transl., Stechert, New York, 1937. · Zbl 0008.15104
[3] Z. Zahorski, Sur la première dérivée, Trans. Amer. Math. Soc. 69 (1950), 1 – 54 (French). · Zbl 0038.20602
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