(************************************************************************)
(*         *   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)         *)
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Local Open Scope R_scope.
(** * Preliminaries lemmas *)
(* end hide *)
(** Intermediate Value Theorem on an Interval (Proof mainly taken from Reals.Rsqrt_def) and its corollary *)
(* begin hide *)
Ltac case_le H :=
let t := type of H in
let h' := fresh in
match t with ?x <= ?y => case (total_order_T x y);
[intros h'; case h'; clear h' |
intros h'; clear -H h'; elimtype False; lra ] end.
(* end hide *)
(** ** The derivative of a reciprocal function       *)
(** * Continuity of the reciprocal function *)

Lemma continuity_pt_recip_prelim : forall (f g:R->R) (lb ub : R) (Pr1:lb < ub),
(forall x y, lb <= x -> x < y -> y <= ub -> f x < f y) ->
(forall x, lb <= x <= ub -> (comp g f) x = id x) ->
(forall a, lb <= a <= ub -> continuity_pt f a) ->
forall b,
f lb < b < f ub ->
continuity_pt g b.