Calculus III: Derivatives and composition

[rman144]

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.

 

[mighty2000]

You are on the right track with your approach. Since you are trying to disprove the statement, you should focus on finding a counterexample that shows the statement to be false. As you have already noted, the key is in the derivative of (g o f)(X). The derivative of a composition of functions is given by the chain rule, and in this case, it should simplify to the identity matrix. Therefore, you can start by looking for functions f and g that would result in a derivative of the identity matrix when composed together. This will help you find a counterexample to the statement. Keep in mind that the functions do not have to be linear, so you may need to think outside the box. Good luck!