(** Continuity and limit *)

Lemma C0_extension_le {T : UniformSpace} (f : R -> T) (a b : R) :
(forall c : R, a <= c <= b -> filterlim f (locally c) (locally (f c))) ->
{g : R -> T | (forall c, filterlim g (locally c) (locally (g c)))
/\ (forall c : R, a <= c <= b -> g c = f c)}.