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.