Abstract
The Cantor ternary function is the most famous example of a continuous function of bounded variation for which it satisfies the Holder continuous function of order α = log3 2, but does not satisfy for order α = 1. In this paper, based on previous work of Hölder continuous function of order α ∈ (0, 1) and using F α - calculus on fractal set F, we show the relation between the Hölder continuous function of order α ∈ (0, 1) and function of bounded variation. In particular, we give the necessary and sufficient condition for the variation of function satisfies Hölder condition or bi-Hölder condition.
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