End continuity.
Arguments continuous {X} {Y}.
Arguments continuous_at {X} {Y}.

(* Specifications: *)
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.
Lemma continuous_composition: forall {X Y Z:TopologicalSpace}
(f:point_set Y -> point_set Z) (g:point_set X -> point_set Y),
continuous f -> continuous g ->
continuous (fun x:point_set X => f (g x)).
Lemma continuous_identity: forall (X:TopologicalSpace),
continuous (fun x:point_set X => x).
Lemma continuous_constant: forall (X Y:TopologicalSpace)
(y0:point_set Y), continuous (fun x:point_set X => y0).
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.