(** * 2D-continuity *)
(** ** Definitions *)
Definition continuity_2d_pt f x y :=
forall eps : posreal, locally_2d (fun u v => Rabs (f u v - f x y) < eps) x y.

Lemma uniform_continuity_2d :
forall f a b c d,
(forall x y, a <= x <= b -> c <= y <= d -> continuity_2d_pt f x y) ->
forall eps : posreal, exists delta : posreal,
forall x y u v,
a <= x <= b -> c <= y <= d ->
a <= u <= b -> c <= v <= d ->
Rabs (u - x) < delta -> Rabs (v - y) < delta ->
Rabs (f u v - f x y) < eps.