(** Continuity and limit *)

Lemma C0_extension_lt {T : UniformSpace} (f : R -> T) la lb (a b : R) :
a < b ->
(forall c : R, a < c < b -> filterlim f (locally c) (locally (f c))) ->
(filterlim f (at_right a) (locally la)) ->
(filterlim f (at_left b) (locally lb)) ->
{g : R -> T | (forall c, filterlim g (locally c) (locally (g c)))
/\ (forall c : R, a < c < b -> g c = f c) /\ g a = la /\ g b = lb}.