(** * Some limits on real functions *)
Definition Rbar_loc_seq (x : Rbar) (n : nat) := match x with
| Finite x => x + / (INR n + 1)
| p_infty => INR n
| m_infty => - INR n
end.

(* Specifications: *)
Lemma filterlim_Rbar_loc_seq :
forall x, filterlim (Rbar_loc_seq x) eventually (Rbar_locally' x).
Lemma continuity_pt_locally :
forall f x,
continuity_pt f x <->
forall eps : posreal, locally x (fun u => Rabs (f u - f x) < eps).
Lemma continuity_pt_locally' :
forall f x,
continuity_pt f x <->
forall eps : posreal, locally' x (fun u => Rabs (f u - f x) < eps).
Lemma continuity_pt_filterlim :
forall (f : R -> R) (x : R),
continuity_pt f x <->
filterlim f (locally x) (locally (f x)).
Lemma continuity_pt_filterlim' :
forall f x,
continuity_pt f x <->
filterlim f (locally' x) (locally (f x)).
Lemma locally_pt_comp (P : R -> Prop) (f : R -> R) (x : R) :
locally (f x) P -> continuity_pt f x ->
locally x (fun x => P (f x)).