I know that the inverse function theorem can be proved for differentiable mappings (not $C^1$) by requiring that $Df(x)$ has everywhere maximum rank (here is the reference https://terrytao.wordpress.com/2011/09/12/the-inverse-function-theorem-for-everywhere-differentiable-maps/).
Knowing this I can easily prove the implicit function theorem for differentiable maps such that $D_yf$ (the differential with respect to the last $n$ variables) has maximum rank.
I was wondering however if I can prove the implicit function theorem for maps that are continuous for the first $m-n$ variables, and differentiable for the last $n$ variables, supposing $D_yf$ with maximum rank (I think i can only prove local existence, local uniqueness and continuity of the implicit function).
The thing is, I couldn't find a reference, hence I was wondering if someone either knew a proof, a reference, or a counterexample.
Thanks in advance!