How to estimate the size of the neighborhoods in the Inverse Function Theorem

Question

Given a function $f:U \subset V\to W$ such that $\textbf{D}f(x_0)\neq 0$ for some $x_0$. How to estimate the neighborhood for which it's invertible? Assuming the second derivative exists and is continuous.

Answer 1

This kind of estimate is rarely needed, but in fact, sometimes one would like to know this.

One estimate of the type you are looking for can be found in Serge Langs Real Analysis (which is, in my opinion, a highly underestimated book on Real analysis), 2nd edition, Chapter 6 §1 Lemma 1.3. It depends on a continuity estimate for $f^\prime$, which you may or may not get from the assumed continuity of the second derivative (depends on what exactly you do know about $d^2 f$...).

Edit this is the same lemma as in Lang's Real and Functional Analysis, Chapter XIV, §1, Lemma 1.3, if you cannot get hold of the Real Analysis.