(** ** Limits, integrals and differentiability *)
Definition is_connected (D : R -> Prop) :=
forall a b x, D a -> D b -> a <= x <= b -> D x.

(* Specifications: *)
Lemma CVU_limits_open (fn : nat -> R -> R) (D : R -> Prop) :
open D
-> CVU_dom fn D
-> (forall x n, D x -> ex_finite_lim (fn n) x)
-> forall x, D x -> ex_finite_lim_seq (fun n => real (Lim (fn n) x))
/\ ex_finite_lim (fun y => real (Lim_seq (fun n => fn n y))) x
/\ real (Lim_seq (fun n => real (Lim (fn n) x)))
= real (Lim (fun y => real (Lim_seq (fun n => fn n y))) x).
Lemma CVU_cont_open (fn : nat -> R -> R) (D : R -> Prop) :
open D ->
CVU_dom fn D ->
(forall n, forall x, D x -> continuity_pt (fn n) x)
-> forall x, D x -> continuity_pt (fun y => real (Lim_seq (fun n => fn n y))) 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))))).