The answer by Nicola Ciccoli is very nice! I looked up Dini's text and the "Inverse Function Theorem" is almost there. I am not talking about topology, just the fact that in the case of a function $f:\mathbb{R}\to\mathbb{R}$ he explicitly says 'derivate delle funzioni inverse', whereas in the whole 50-page Chapter XIII, he never uses the word inverse. Of course, it is a simple corollary of the Implicit Function Theorem. At some point in time this changed. Now most of the texts prove Inverse Function Theorem first, then derive Implicit Function Theorem. When did that start? Who was the first?