apply continuity_implies_ex_Rint.
apply Cu1.
Qed.
Section Beta.
Definition beta (x t : R) := 1/(2*c) * RInt (fun u => u1 u) (x - c * t) (x + c * t).
Definition beta20 x t := 1/(2*c) * (Derive (fun u => u1 u) (x + c * t) - Derive (fun u => u1 u) (x - c * t)).
Definition beta01 x t := 1/2 * (u1 (x + c * t) + u1 (x - c * t)).
Definition beta02 x t := c/2 * (Derive (fun u => u1 u) (x + c * t) - Derive (fun u => u1 u) (x - c * t)).

(* Specifications: *)
Lemma beta20_lim :
forall x t, is_derive_n (fun u => beta u t) 2 x (beta20 x t).
Lemma beta01_lim :
forall x t, is_derive (fun u => beta x u) t (beta01 x t).
Lemma beta02_lim :
forall x t, is_derive_n (fun u => beta x u) 2 t (beta02 x t).