(** * 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.

Lemma continuity_pt_locally' :
forall f x,
continuity_pt f x <->
forall eps : posreal, locally' x (fun u => Rabs (f u - f x) < eps).