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.

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.