(** ** in [R] *)

(* Specifications: *)
Lemma continuous_Rabs (x : R) :
continuous Rabs x.
Lemma filterlim_Rabs (x : Rbar) :
filterlim Rabs (Rbar_locally' x) (Rbar_locally (Rbar_abs x)).
Lemma is_lim_Rabs (f : R -> R) (x l : Rbar) :
is_lim f x l -> is_lim (fun x => Rabs (f x)) x (Rbar_abs l).
Lemma filterlim_Rabs_0 :
filterlim Rabs (Rbar_locally' 0) (at_right 0).
Lemma is_lim_Rabs_0 (f : R -> R) (x : Rbar) :
is_lim f x 0 -> Rbar_locally' x (fun x => f x <> 0)
-> filterlim (fun x => Rabs (f x)) (Rbar_locally' x) (at_right 0).
Lemma filterdiff_Rabs (x : R) :
x <> 0 -> filterdiff Rabs (locally x) (fun y : R => scal y (sign x)).
Lemma is_derive_Rabs (f : R -> R) (x df : R) :
is_derive f x df -> f x <> 0
-> is_derive (fun x => Rabs (f x)) x (sign (f x) * df).