(* 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'; exfalso; lra ] end.

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}.