Definition image_dir (f:R -> R) (D:R -> Prop) (x:R) : Prop :=
  exists y : R, x = f y /\ D y.

Lemma prolongement_C0 : forall (f:R -> R) (a b:R), a <= b -> (forall c:R, a <= c <= b -> continuity_pt f c) -> exists g : R -> R, continuity g /\ (forall c:R, a <= c <= b -> g c = f c).