Approximations of nondifferentiable functions and local topological equivalence

A concept of local approximation of a function is introduced. This concept is defined via directional derivatives. In consequence, the local approximation is carried out by a positively homogeneous mapping. We obtain local approximations for functions that are not necessarily locally Lipschitzian nor continuous. This is the case of some large classes of functions such as stable functions or contingently epidifferentiable and directionally Lipschitzian functions. Using the concept of topological equivalence we establish the existence of a local coordinate transformation between the original function and the positively homogeneous function. This investigation is developed for contingently epidifferentiable functions around a noncritical point, and for noncontingently epidifferentiable functions under particular conditions.