Please, can you give me some hints about this? Here is a proof of the inverse function theorem. 1. After the statement and proof of a previous lemma, the author puts (L o f) as a composite function. I don't understand this because L is a matrix (the jacobian matrix of f(a) ) and I have not seen in my book (Apostol's) that one can directly consider a matrix as a function that may be articulated with other function to build a composite one. 2. In the proof of Claim 1 the author puts this (j and i are subindexes): l Dj gi (x) l = l Dj fi (x) - Dj fi (a) l and I would thank if you can tell me how he introduces the Dj fi (a) there (because, following the definition of function g, I thought that Dj gi (x) = Dj fi (x) ). Thanks for your good will and your time.