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Characterization of the Local Growth of Two Cantor-Type Functions

1
Environment, Health and Safety, IMEC vzw, Kapeldreef 75, 3001 Leuven, Belgium
2
Neuroscience Research Flanders, 3001 Leuven, Belgium
Fractal Fract 2019, 3(3), 45; https://doi.org/10.3390/fractalfract3030045
Received: 21 June 2019 / Revised: 30 July 2019 / Accepted: 6 August 2019 / Published: 21 August 2019
The Cantor set and its homonymous function have been frequently utilized as examples for various physical phenomena occurring on discontinuous sets. This article characterizes the local growth of the Cantor’s singular function by means of its fractional velocity. It is demonstrated that the Cantor function has finite one-sided velocities, which are non-zero of the set of change of the function. In addition, a related singular function based on the Smith–Volterra–Cantor set is constructed. Its growth is characterized by one-sided derivatives. It is demonstrated that the continuity set of its derivative has a positive Lebesgue measure of 1/2. View Full-Text
Keywords: singular functions; Hölder classes; differentiability; fractional velocity; Smith–Volterra–Cantor set singular functions; Hölder classes; differentiability; fractional velocity; Smith–Volterra–Cantor set
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Prodanov, D. Characterization of the Local Growth of Two Cantor-Type Functions. Fractal Fract 2019, 3, 45.

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