(** * Definitions on [R] *)

Definition Derive (f : R -> R) (x : R) := real (Lim (fun h => (f (x+h) - f x)/h) 0).

Lemma incr_function_le (f : R -> R) (a b : Rbar) (df : R -> R) : (forall (x : R), Rbar_le a x -> Rbar_le x b -> is_derive f x (df x)) -> ((forall (x : R), Rbar_le a x -> Rbar_le x b -> df x > 0) -> (forall (x y : R), Rbar_le a x -> x < y -> Rbar_le y b -> f x < f y)).