Definition pos_div_2 (eps : posreal) := mkposreal _ (is_pos_div_2 eps).(** * The sign function *)

Definition sign (x : R) :=
  match total_order_T 0 x with
  | inleft (left _) => 1
  | inleft (right _) => 0
  | inright _ => -1
  end.

Lemma Riemann_integrable_bound (f : R -> R) (a b : R) : Riemann_integrable f a b -> exists s : R, forall x, Rmin a b <= x <= Rmax a b -> f x <= s.