(************************************************************************)
(*         *   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 *)

Lemma f_interv_is_interv : forall (f:R->R) (lb ub y:R),
lb < ub ->
f lb <= y <= f ub ->
(forall x, lb <=  x <= ub -> continuity_pt f x) ->
{x | lb <= x <= ub /\ f x = y}.