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Calculus III: Derivatives and composition

  1. Oct 19, 2009 #1
    I need prove or disprove the following statement:

    If f: R^2->R^2 is a differentiable function whose derivative at (0,0) is not invertible, then there does not exist a differentiable function g: R^2->R^2 such that:

    (g o f)(X)=X

    I've been trying to find counter examples like crazy (f(x,y)=(x+y,x-y), (xy,x+y), etc.) but everything seems to agree. I think this is actually true, but I don't want an entire proof (I can do that work); just a hint or two as to which way I should begin to approach this (basically I just need a starting point).

    Nevermind; I found the trick. I realized if (g o f)(X)=X, the derivative of this must be the identity matrix. From there, no problems.
    Last edited: Oct 19, 2009
  2. jcsd
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