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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.