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_neighborhood_basis: forall (x:point_set X) (NB:Family (point_set Y)), neighborhood_basis NB (f x) -> (forall V:Ensemble (point_set Y), In NB V -> neighborhood (inverse_image f V) x) -> continuous_at x.