destruct (Du1 x) as (l,Hl).
apply derivable_continuous_pt.
unfold derivable_pt, derivable_pt_abs.
exists l.
now apply is_derive_Reals.
Qed.

Lemma continuity_implies_ex_Rint: forall f a b,
(forall x, continuity_pt f x) -> ex_RInt f a b.