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).

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