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_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.