Prove that the differentiable function can never have a differentiable inverse.

My attempt : I notice that and that . Hence is not injective, hence doesn't have an inverse, so it cannot have a differentiable inverse.

I'm not sure my attempt is good.

Maybe I should have done it using the Inverse Function Theorem, but if the Jacobian of is not invertible, I can't say anything whether the existence of an inverse of , I believe. Hmm but according to my memory of Linear Algebra, if the matrix of a function is not invertible then the function is not injective so it doesn't have an inverse.

What do you think?