Fixpoint SubEquiN (N:nat) (x y:R) (del:posreal) : list R :=
  match N with
    | O => cons y nil
    | S p => cons x (SubEquiN p (x + del) y del)
  end.Definition max_N (a b:R) (del:posreal) (h:a < b) : nat :=
  let (N,_) := maxN del h in N.Definition SubEqui (a b:R) (del:posreal) (h:a < b) : list R :=
  SubEquiN (S (max_N del h)) a b del.

Lemma continuity_implies_RiemannInt : forall (f:R -> R) (a b:R), a <= b -> (forall x:R, a <= x <= b -> continuity_pt f x) -> Riemann_integrable f a b.