Summary: Weierstrass’s everywhere continuous but nowhere differentiable function is shown to be locally continuously fractionally differentiable everywhere for all orders below the “critical 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 differentiability and the box dimension/local Hölder exponent. Lévy index for one dimensional Lévy flights 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 Hölder exponent. It is argued that local fractional derivatives provide a powerful tool to analyze pointwise behavior of irregular signals.