Lemma null_derivative_loc : forall (f:R -> R) (a b:R) (pr:forall x:R, a < x < b -> derivable_pt f x), (forall x:R, a <= x <= b -> continuity_pt f x) -> (forall (x:R) (P:a < x < b), derive_pt f x (pr x P) = 0) -> constant_D_eq f (fun x:R => a <= x <= b) (f a).