(** * 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 x c) ->
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 u v - f x y) < eps.