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This concerns the convergence of Newton's method in unconstrained optimization. But the question can be taken out of this context.

Suppose $f$ is twice differentiable and that the Hessian $\triangledown^2f(x)$ is Lipschitz continuous in a neighborhood of $x^*$, and $\triangledown^2f(x)$ is invertible at $x=x^*$. Questions are

  1. Is there a neighborhood of $x^*$ where $\triangledown^2f(x)$ is invertible?
  2. Is there a neighborhood of $x^*$ where $||\triangledown^2f(x)^{-1}||_2\leq2||\triangledown^2f(x^*)^{-1}||_2$

I've taken one semester of Analysis and Numerical Linear Algebra, but couldn't crack this. If you know the answer to the question, could you also suggest a book to read where this sort of subject is covered?

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  • $\begingroup$ Determinant is a smooth function. Inversion is a rational function, which is smooth on its domain. $\endgroup$ – user251257 Jan 30 '18 at 3:07

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