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).
End two_arg_convenience_results.
Arguments continuous_2arg {X} {Y} {Z}.
Arguments continuous_at_2arg {X} {Y} {Z}.

Corollary continuous_composition_2arg:
forall {U X Y Z : TopologicalSpace} (f : U -> X) (g : U -> Y) (h : X -> Y -> Z),
continuous f -> continuous g -> continuous_2arg h ->
continuous (fun p => h (f p) (g p)).