(** ** Limits, integrals and differentiability *)

Definition is_connected (D : R -> Prop) :=
  forall a b x, D a -> D b -> a <= x <= b -> D x.

Lemma CVU_Derive (fn : nat -> R -> R) (D : R -> Prop) : open D -> is_connected D -> CVU_dom fn D -> (forall n x, D x -> ex_derive (fn n) x) -> (forall n x, D x -> continuity_pt (Derive (fn n)) x) -> CVU_dom (fun n x => Derive (fn n) x) D -> (forall x , D x -> (is_derive (fun y => real (Lim_seq (fun n => fn n y))) x (real (Lim_seq (fun n => Derive (fn n) x))))).