Corollary diff_continuous: forall (X:TopologicalSpace)
  (f g:point_set X -> point_set RTop) (x:point_set X),
  continuous_at f x -> continuous_at g x ->
  continuous_at (fun x:point_set X => f x - g x) x (Y:=RTop).

Lemma const_multiple_func_continuous: forall c:R, continuous (fun x:R => c*x) (X:=RTop) (Y:=RTop).