  We introduce the scale calculus, which generalizes the classical differential
calculus to non differentiable functions. The new derivative is called the
scale difference operator. We also introduce the notions of fractal functions,
minimal resolution, and quantum representation of a non differentiable
function. We then define a scale quantization procedure for classical
Lagrangian systems inspired by the Scale relativity theory developped by
Nottale. We prove that the scale quantization of Newtionian mechanics is a non
linear Schrodinger equation. Under some specific assumptions, we obtain the
classical linear Schrodinger equation.
