From the result discussed in Does the inverse function theorem hold for everywhere differentiable maps? (which I'll call the differentiable nonsmooth Inverse Function Theorem) one can obtain a differentiable but nonsmooth version of the Implicit Function Theorem by the usual argument.
There are two interesting closely related questions apparently still remaining:
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Is there a corresponding version of the continuous Implicit Function Theorem not requiring continuous differentiability in the variables being solved for?
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In the original inductive proof of the Implicit Function Theorem, continuous differentiability was needed to insure a decreasing chain of locally nonvanishing minors for the Jacobian determinant. Does there always exist such a chain if simple differentiability with nonzero Jacobian is assumed?
A positive answer to 2 gives a positive answer to 1 and an inductive proof of the differentiable nonsmooth Inverse Function Theorem.
For a more careful description of these problems, see the exposition of the Implicit Function Theorem in my real analysis manuscript on my website http://wolfweb.unr.edu/homepage/bruceb/ .
Best Answer
If I remember correctly, strict differentiability at one point $x_0$ (and of course invertibility of $df(x_0)$) is sufficient for the implicit function theorem to hold. This means something like $$ df(x_0)(h) = lim_{t\to 0, x\to x_0}\frac1t f(x+th). $$ I am traveling and I do not have access to the books now. This is in:
MR0724435 Ver Eecke, Paul Fondements du calcul différentiel. (French) [Foundations of differential calculus] Mathématiques. [Mathematics] Presses Universitaires de France, Paris, 1983. 345 pp. ISBN: 2-13-038180-4 (Reviewer: William Eames) 26-01 (46G05 58-01 58C20)
MR0817719 (87e:58001) Ver Eecke, Paul(F-PCRD) Applications du calcul différentiel. (French) [Applications of differential calculus] Mathématiques. [Mathematics] Presses Universitaires de France, Paris, 1985. 397 pp. ISBN: 2-13-038961-9 58-01 (26E15)