Definition extension_C0 (f : R -> V) (a b : Rbar) (x : R) : V :=
  match Rbar_le_dec a x with
    | left _ => match Rbar_le_dec x b with
        | left _ => f x
        | right _ => f (real b)
      end
    | right _ => f (real a)
  end.

Lemma extension_C0_continuous (f : R -> V) (a b : Rbar) : Rbar_le a b -> (forall x : R, Rbar_le a x -> Rbar_le x b -> continuous f x) -> forall x, continuous (extension_C0 f a b) x.