A metric approach to asymptotic analysis

We introduce a notion of “firm” (or uniform) asymptotic cone to an unbounded subset of a normed space. We relate this notion to a concept of “firm” asymptotic function. We use these notions to study boundedness properties which can be applied to continuity questions for some operations on sets and functions. Such questions arise in stability analysis of Hamilton-Jacobi equations. We present some other applications such as an extension of a theorem of Dieudonné and existence results in optimization and fixed point theory.