(* Ropp_continuous was already proved in RTopology *)

(* Specifications: *)
Lemma Rminus_continuous: continuous_2arg Rminus
(X:=RTop) (Y:=RTop) (Z:=RTop).
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).
Corollary const_multiple_continuous: forall (X:TopologicalSpace)
(f:point_set X -> point_set RTop) (c:R) (x:point_set X),
continuous_at f x -> continuous_at (fun x:point_set X => c * f x) x
(Y:=RTop).
Lemma Rmult_continuous_at_origin: continuous_at_2arg Rmult 0 0
(X:=RTop) (Y:=RTop) (Z:=RTop).
Lemma Rmult_continuous: continuous_2arg Rmult (X:=RTop) (Y:=RTop) (Z:=RTop).
Corollary product_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 Rinv_continuous_at_1: continuous_at Rinv 1 (X:=RTop) (Y:=RTop).
Lemma Rinv_continuous: forall x0:R, x0<>0 -> continuous_at Rinv x0
(X:=RTop) (Y:=RTop).
Lemma Rdiv_continuous: forall x y:R, y <> 0 ->
continuous_at_2arg Rdiv x y (X:=RTop) (Y:=RTop) (Z:=RTop).
Corollary quotient_continuous: forall (X:TopologicalSpace)
(f g:point_set X -> point_set RTop) (x0:point_set X),
continuous_at f x0 -> continuous_at g x0 -> g x0 <> 0 ->
continuous_at (fun x:point_set X => f x / g x) x0 (Y:=RTop).
Lemma Rabs_continuous: continuous Rabs (X:=RTop) (Y:=RTop).
Require Export Homeomorphisms.