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

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.