Implicit Type f : R -> R.(****************************************************)
(** *         Basic operations on functions         *)
(****************************************************)
Definition plus_fct f1 f2 (x:R) : R := f1 x + f2 x.Definition opp_fct f (x:R) : R := - f x.Definition mult_fct f1 f2 (x:R) : R := f1 x * f2 x.Definition mult_real_fct (a:R) f (x:R) : R := a * f x.Definition minus_fct f1 f2 (x:R) : R := f1 x - f2 x.Definition div_fct f1 f2 (x:R) : R := f1 x / f2 x.Definition div_real_fct (a:R) f (x:R) : R := a / f x.Definition comp f1 f2 (x:R) : R := f1 (f2 x).Definition inv_fct f (x:R) : R := / f x.Definition mirr_fct f (x:R) : R := f (- x).Infix "+" := plus_fct : Rfun_scope.Notation "- x" := (opp_fct x) : Rfun_scope.Infix "*" := mult_fct : Rfun_scope.Infix "-" := minus_fct : Rfun_scope.Infix "/" := div_fct : Rfun_scope.Local Notation "f1 'o' f2" := (comp f1 f2)
  (at level 20, right associativity) : Rfun_scope.Notation "/ x" := (inv_fct x) : Rfun_scope.Definition fct_cte (a x:R) : R := a.Definition id (x:R) := x.(****************************************************)
(** *          Variations of functions              *)
(****************************************************)
Definition increasing f : Prop := forall x y:R, x <= y -> f x <= f y.Definition decreasing f : Prop := forall x y:R, x <= y -> f y <= f x.Definition strict_increasing f : Prop := forall x y:R, x < y -> f x < f y.Definition strict_decreasing f : Prop := forall x y:R, x < y -> f y < f x.Definition constant f : Prop := forall x y:R, f x = f y.(**********)
Definition no_cond (x:R) : Prop := True.(**********)
Definition constant_D_eq f (D:R -> Prop) (c:R) : Prop :=
  forall x:R, D x -> f x = c.(***************************************************)
(** *    Definition of continuity as a limit       *)
(***************************************************)

(**********)
Definition continuity_pt f (x0:R) : Prop := continue_in f no_cond x0.Definition continuity f : Prop := forall x:R, continuity_pt f x.

Lemma continuity_pt_comp : forall f1 f2 (x:R), continuity_pt f1 x -> continuity_pt f2 (f1 x) -> continuity_pt (f2 o f1) x.