(** ** 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.