Fractional Diierentiability of Nowhere Diierentiable Functions and Dimensions

Weierstrass's everywhere continuous but nowhere diierentiable function is shown to be locally continuously fractionally diierentiable everywhere for all orders below thècritical order' 2?s and not so for orders between 2?s and 1, where s, 1 < s < 2, is the box dimension of the graph of the function. This observation is consolidated in the general result showing a direct connection between local fractional diierentiability and the box dimension/ local HH older exponent. L evy index for one dimensional L evy ights is shown to be the critical order of its characteristic function. Local fractional derivatives of multifractal signals (non-random functions) are shown to provide the local HH older exponent. It is argued that Local fractional derivatives provide a powerful tool to analyze pointwise behavior of irregular signals.