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).