(** ** 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_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.