Definition extension_cont (f g : R -> U) (a x : R) : U :=
  match Rle_dec x a with
    | left _ => f x
    | right _ => g x
  end.

(* Specifications: *)
Lemma extension_cont_continuous (f g : R -> U) (a : R) : continuous f a -> continuous g a -> f a = g a -> continuous (extension_cont f g a) a.
Lemma extension_cont_is_derive (f g : R -> V) (a : R) (l : V) : is_derive f a l -> is_derive g a l -> f a = g a -> is_derive (extension_cont f g a) a l.