Require Export Continuity.

Lemma metric_space_fun_continuity: forall (X Y:TopologicalSpace)
(f:point_set X->point_set Y) (x:point_set X)
(dX:point_set X -> point_set X -> R)
(dY:point_set Y -> point_set Y -> R),
metrizes X dX -> metrizes Y dY ->
(forall eps:R, eps > 0 -> exists delta:R, delta > 0 /\
forall x':point_set X, dX x x' < delta ->
dY (f x) (f x') < eps) ->
continuous_at f x.