(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
Local Open Scope R_scope.
(* Boule is French for Ball *)
Definition Boule (x:R) (r:posreal) (y:R) : Prop := Rabs (y - x) < r.
(* General properties of balls. *)
Definition boule_of_interval x y (h : x < y) :
{c :R & {r : posreal | c - r = x /\ c + r = y}}.
Definition boule_in_interval x y z (h : x < z < y) :
{c : R & {r | Boule c r z /\  x < c - r /\ c + r < y}}.
(** Uniform convergence *)
Definition CVU (fn:nat -> R -> R) (f:R -> R) (x:R)
(r:posreal) : Prop :=
forall eps:R,
0 < eps ->
exists N : nat,
(forall (n:nat) (y:R),
(N <= n)%nat -> Boule x r y -> Rabs (f y - fn n y) < eps).
(** Normal convergence *)
Definition CVN_r (fn:nat -> R -> R) (r:posreal) : Type :=
{ An:nat -> R &
{ l:R |
Un_cv (fun n:nat => sum_f_R0 (fun k:nat => Rabs (An k)) n) l /\
(forall (n:nat) (y:R), Boule 0 r y -> Rabs (fn n y) <= An n) } }.
Definition CVN_R (fn:nat -> R -> R) : Type := forall r:posreal, CVN_r fn r.
Definition SFL (fn:nat -> R -> R)
(cv:forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l })
(y:R) : R := let (a,_) := cv y in a.

(** Each limit of a sequence of functions which converges uniformly is continue *)
Lemma CVU_continuity :
forall (fn:nat -> R -> R) (f:R -> R) (x:R) (r:posreal),
CVU fn f x r ->
(forall (n:nat) (y:R), Boule x r y -> continuity_pt (fn n) y) ->
forall y:R, Boule x r y -> continuity_pt f y.