Continuous approximations of nonsmooth mappings

We introduce the concept of approximator, i.e. a first order local approximation of a mapping, which depends on some point and on the direction without any assumptions of homogeneity. This is done in order to overcome the tight connection, shown by the directional derivative, between the continuity with respect to the point and linearity with respect to the direction. We obtain in this way an exact approximation of nonsmooth mappings which depends continuously on the point. Various examples of mappings admitting continuous approximators are given and some known results of classical analysis are extended to this nondifferentiable setting. Moreover we show how the theoretical machinery can be applied in various fields such as fixed point theory, optimization theory, and Newton's methods.