(* 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.
(* 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}.