The Fitzpatrick Rate, the Stepanov Rate, the Lipschitz Rate and their Relatives

We give some variants of a constant associated to a real-valued function on a Banach space and a point of the space, as introduced by S. Fitzpatrick. Some of them have a one-sided character and enable to give criteria of subdifferentiability (or differentiability) of different types. An application to best approximation is presented.