Elements for a metric tangential calculus

The metric jets, introduced in the first chapter, generalize the jets (at order one) of Charles Ehresmann. In short, for a “good” map $f$ (said to be “tangentiable” at $a$), we define its metric jet tangent at $a$ (composed of all the maps which are locally lipschitzian at $a$ and tangent to $f$ at $a$) called the “tangential” of $f$ at $a$, and denoted T$f_a$ (the domain and codomain of $f$ being metric spaces). Furthermore, guided by the heuristic example of the metric jet T$f_a$, tangent to a map $f$ differentiable at $a$, which can be canonically represented by the unique continuous affine map it contains, we will extend, in the second chapter, into a specific metric context, this property of representation of a metric jet.This yields a lot of relevant examples of such representations.