About non-differentiable functions

We develop the notion of local fractional derivative introduced by Kolvankar and Gangal. It allows a fine study of the local structure of irregular (fractal) functions. Using this tool, we extend classical theorems of analysis (extrema, Rolle) to non-differentiable functions. In particular, we prove a generalized Taylor expansion theorem. We introduce a new derivative of real order and discuss its properties. (C) 2001 Academic Press.