(** 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 describes basic properties (such as limits or differentiability) about basic functions: absolute value, inverse, square root, power, exponential and so on.*)
(** * Absolute value *)
(** ** in an [AbsRing] *)
(** ** in [R] *)

Lemma is_lim_Rabs_0 (f : R -> R) (x : Rbar) :
is_lim f x 0 -> Rbar_locally' x (fun x => f x <> 0)
-> filterlim (fun x => Rabs (f x)) (Rbar_locally' x) (at_right 0).