(** * 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.