(** ** The derivative of a reciprocal function       *)
(** * Continuity of the reciprocal function *)

(* Specifications: *)
Lemma continuity_pt_recip_prelim : forall (f g:R->R) (lb ub : R) (Pr1:lb < ub),
(forall x y, lb <= x -> x < y -> y <= ub -> f x < f y) ->
(forall x, lb <= x <= ub -> (comp g f) x = id x) ->
(forall a, lb <= a <= ub -> continuity_pt f a) ->
forall b,
f lb < b < f ub ->
continuity_pt g b.
Lemma continuity_pt_recip_interv : forall (f g:R->R) (lb ub : R) (Pr1:lb < ub),
(forall x y, lb <= x -> x < y -> y <= ub -> f x < f y) ->
(forall x, f lb <= x -> x <= f ub -> (comp f g) x = id x) ->
(forall x, f lb <= x -> x <= f ub -> lb <= g x <= ub) ->
(forall a, lb <= a <= ub -> continuity_pt f a) ->
forall b,
f lb < b < f ub ->
continuity_pt g b.