  In this note, we discuss a generalization of the well-known implicit function
theorem to the time-delay case. We show that the latter problem is closely
related to the bicausal changes of coordinates of time-delay systems. An
iterative algorithm is proposed to check the conditions and to construct the
desired bicausal change of coordinates for the proposed implicit function
theorem. Moreover, we show that our results can be applied to delayed
differential-algebraic equations (DDAEs) to reduce their indices and to get
their solutions. Some numerical examples are given to illustrate our results.
