A Short Note on Local Fractional Calculus of Function of One Variable

Yang Xiao-Jun

Abstract


Local fractional calculus (LFC) handles everywhere continuous but nowhere differentiable functions in fractal space. This note investigates the theory of local fractional derivative and integral of function of one variable. We first introduce the theory of local fractional continuity of function and history of local fractional calculus. We then consider the basic theory of local fractional derivative and integral, containing the local fractional Rolle’s theorem, L’Hospital’s rule, mean value theorem, anti-differentiation and related theorems, integration by parts and Taylor’ theorem. Finally, we study the efficient application of local fractional derivative to local fractional extreme value of non-differentiable functions, and give the theory of local fractional extreme value, containing local fractional extreme value theorem, Fermat’s theorem, increasing/decreasing test and the derivative test. It is of great significances for us to process optimization problems of the non-differentiable functions on Cantor set. 


Keywords


Local fractional calculus; Local fractional Taylor theorem; Mean value theorem; Local fractional Fermat’s theorem; Local extreme value theorem; Cantor set

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