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

(* Specifications: *)
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Lemma continuity_compact :
forall (f:R -> R) (X:R -> Prop),
(forall x:R, continuity_pt f x) -> compact X -> compact (image_dir f X).
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).
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Lemma continuity_ab_maj :
forall (f:R -> R) (a b:R),
a <= b ->
(forall c:R, a <= c <= b -> continuity_pt f c) ->
exists Mx : R, (forall c:R, a <= c <= b -> f c <= f Mx) /\ a <= Mx <= b.
(**********)
Lemma continuity_ab_min :
forall (f:R -> R) (a b:R),
a <= b ->
(forall c:R, a <= c <= b -> continuity_pt f c) ->
exists mx : R, (forall c:R, a <= c <= b -> f mx <= f c) /\ a <= mx <= b.