End product_topology2.
Section two_arg_convenience_results.
Variable X Y Z:TopologicalSpace.
Variable f:point_set X -> point_set Y -> point_set Z.
Definition continuous_2arg :=
continuous (fun p:point_set X * point_set Y =>
let (x,y):=p in f x y)
(X:=ProductTopology2 X Y).
Definition continuous_at_2arg (x:point_set X) (y:point_set Y) :=
continuous_at (fun p:point_set X * point_set Y =>
let (x,y):=p in f x y)  (x, y)
(X:=ProductTopology2 X Y).

(* Specifications: *)
Lemma continuous_2arg_func_continuous_everywhere:
continuous_2arg -> forall (x:point_set X) (y:point_set Y),
continuous_at_2arg x y.
Lemma pointwise_continuity_2arg:
(forall (x:point_set X) (y:point_set Y),
continuous_at_2arg x y) -> continuous_2arg.