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_composition_at: forall {X Y Z:TopologicalSpace} (f:point_set Y -> point_set Z) (g:point_set X -> point_set Y) (x:point_set X), continuous_at f (g x) -> continuous_at g x -> continuous_at (fun x:point_set X => f (g x)) x.