(*********)
Definition D_x (D:R -> Prop) (y x:R) : Prop := D x /\ y <> x.(*********)
Definition continue_in (f:R -> R) (D:R -> Prop) (x0:R) : Prop :=
  limit1_in f (D_x D x0) (f x0) x0.(*********)
Definition D_in (f d:R -> R) (D:R -> Prop) (x0:R) : Prop :=
  limit1_in (fun x:R => (f x - f x0) / (x - x0)) (D_x D x0) (d x0) x0.

Lemma Dx_pow_n : forall (n:nat) (D:R -> Prop) (x0:R), D_in (fun x:R => x ^ n) (fun x:R => INR n * x ^ (n - 1)) D x0.