  Consider a Hausdorff space (X,T) and a set C of converging nets in X. By
virtue of the limit uniqueness, the relation Lim which assigns each member x of
X to every net N lying in C that converges to x is a map. Of course,
structuring C with some topology U, Lim can be a continuous map. If T is a
topology induced by a uniformity, and F is a function space such that X is the
codomain of each f in F, it is a well-known property the uniform limits of
continuous functions to be continuous. In this paper, the author shows that the
continuity of limits of continuous-map nets is implied by the continuity of the
map Lim, whenever the involved topologies are large enough, being this result
obtained without using neither the uniform convergence notion nor the
uniformity concept.
