This paper presents an extended formulation of the basic continuation problem for implicitly defined, embedded manifolds in $Rn$. The formulation is chosen so as to allow for the arbitrary imposition of additional constraints during continuation and the restriction to selective parametrizations of the corresponding higher-codimension solution manifolds. In particular, the formalism is demonstrated to clearly separate between the essential functionality required of core routines in application-oriented continuation packages, on the one hand, and the functionality provided by auxiliary toolboxes that encode classes of continuation problems and user definitions that narrowly focus on a particular problem implementation, on the other hand. Several examples are chosen to illustrate the formalism and its implementation in the recently developed continuation core package COCO and auxiliary toolboxes, including the continuation of families of periodic orbits in a hybrid dynamical system with impacts and friction as well as the detection and constrained continuation of selected degeneracies characteristic of such systems, such as grazing and switching-sliding bifurcations.