(** This file is part of the Coquelicot formalization of real analysis in Coq: http://coquelicot.saclay.inria.fr/  Copyright (C) 2011-2015 Sylvie Boldo #<br /># Copyright (C) 2011-2015 Catherine Lelay #<br /># Copyright (C) 2011-2015 Guillaume Melquiond  This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.  This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the COPYING file for more details. *)
(** This file contains results about the integral as a function: continuity, differentiability, and composition. Theorems on parametric integrals are also provided. *)
(** * Continuity *)
Section Continuity.
Context {V : NormedModule R_AbsRing}.

(* Specifications: *)
Lemma continuous_RInt_0 :
forall (f : R -> V) (a : R) If,
locally a (fun x => is_RInt f a x (If x))
-> continuous If a.
Lemma continuous_RInt_1 (f : R -> V) (a b : R) (If : R -> V) :
locally b (fun z : R => is_RInt f a z (If z))
-> continuous If b.
Lemma continuous_RInt_2 (f : R -> V) (a b : R) (If : R -> V) :
locally a (fun z : R => is_RInt f z b (If z))
-> continuous If a.
Lemma continuous_RInt (f : R -> V) (a b : R) (If : R -> R -> V) :
locally (a,b) (fun z : R * R => is_RInt f (fst z) (snd z) (If (fst z) (snd z)))
-> continuous (fun z : R * R => If (fst z) (snd z)) (a,b).