Uniformity and the delta method

When are asymptotic approximations using the delta-method uniformly valid? We provide sufficient conditions as well as closely related necessary conditions for uniform negligibility of the remainder of such approximations. These conditions are easily verified by empirical practitioners and permit to identify settings and parameter regions where pointwise asymptotic approximations perform poorly. Our framework allows for a unified and transparent discussion of uniformity issues in various sub-fields of statistics and econometrics. Our conditions involve uniform bounds on the remainder of a first-order approximation for the function of interest.