(** * 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_1d' : forall f a b c, (forall x, a <= x <= b -> continuity_2d_pt f c x) -> forall eps : posreal, exists delta : posreal, forall x y u v, a <= x <= b -> c - delta <= y <= c + delta -> a <= u <= b -> c - delta <= v <= c + delta -> Rabs (u - x) < delta -> Rabs (f v u - f y x) < eps.