  Let (X,d) be a metric space and $ \alpha > 0 $. In this paper, we study
extensions of some complex-valued Lipschitz functions, from some special subset
$ X_0 $ to X. These extensions are with no-increasing Lipschitz number or the
smallest Lipschitz number. Moreover, we show that under some conditions, Tietze
extension theorem can be generalized for Lipschitz functions and call it
Tietze-Lipschitz extension. Furthermore, we generalize Urysohn-lemma for
Lipschitz functions. In fact we present a necessary and sufficient condition
for that Lipschitz functions separate subsets of X.
