Section Alpha.
Definition alpha x t := 1/2 * (u0 (x + c * t) + u0 (x - c * t)).
Definition alpha20 x t := 1/2 * (Derive_n u0 2 (x + c * t) + Derive_n u0 2 (x - c * t)).
Definition alpha02 x t := c^2/2 * (Derive_n u0 2 (x + c * t) + Derive_n u0 2 (x - c * t)).
End Alpha.
Parameter u1 : R -> R.
Hypothesis Du1 : forall x, ex_derive (fun u => u1 u) x.
intros x.
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.