generalcontinues

Generalcontinues is a neologism that appears in some mathematical writings to describe a generalized notion of continuity that applies to families of objects indexed by a parameter. Unlike standard continuity, which is defined for a single space and a single map, generalcontinues aims to describe how a collection of maps or structures changes as the index varies, while preserving a chosen sense of closeness across the family.

Informally, a family {f_t} of maps f_t: X_t -> Y_t indexed by t in T is generally continuous if the map (t, x) -> f_t(x) is continuous when the disjoint union of the X_t and Y_t is given an appropriate topology (or when the space of maps carries a suitable topology). This broad idea can be specialized in different contexts by selecting the ambient category and the topology used to measure convergence.

In practice, formulations of generalcontinues differ: some require continuity for all convergent nets of indices, others require uniform-type control over the parameter. The precise definition depends on the chosen setting (topological spaces, manifolds, or more general spaces) and the intended notion of convergence or closeness across the index.

Applications are discussed in parameter-dependent analysis, dynamical systems, and certain category-theoretic contexts where one studies how morphisms vary with a parameter. An example is the family of linear maps f_a: R -> R defined by f_a(x) = a x, which is generally continuous in a when the map (a, x) -> a x is continuous on R × R.

Because generalcontinues is not standardized, readers may encounter different formulations in different texts, and some authors prefer terms such as parameter-continuity or fiberwise continuity. See also continuity, parametric continuity, and continuity in category theory.