Section continuity.
Variable X Y:TopologicalSpace.
Variable f:point_set X -> point_set Y.
Definition continuous : Prop :=
forall V:Ensemble (point_set Y), open V ->
open (inverse_image f V).
Definition continuous_at (x:point_set X) : Prop :=
forall V:Ensemble (point_set Y),
neighborhood V (f x) -> neighborhood (inverse_image f V) x.
End continuity.
Arguments continuous {X} {Y}.
Arguments continuous_at {X} {Y}.

Lemma continuous_at_is_local: forall (X Y:TopologicalSpace)
(x0:point_set X) (f g:point_set X -> point_set Y)
(N:Ensemble (point_set X)),
neighborhood N x0 -> (forall x:point_set X, In N x -> f x = g x) ->
continuous_at f x0 -> continuous_at g x0.