A set mapping $F:X \rightrightarrows Y$ is said to be metrically regular for $\overline{x}\in X$ and $\overline{y} \in Y$ if there exists a $\kappa\in(0,\infty)$ for which $$ d(x,F^{-1}(y))\leq \kappa d(y,F(x)) $$ for all $(x,y)\in X\times Y$ close to $(\overline{x},\overline{y})$.
The regularity modulus is defined as the infimum over all $\kappa$ for which the above inequality holds.
I was wondering what the purpose of the regularity modulus is. If you're taking the infimum over arbitrary neighbourhoods of $(\overline{x},\overline{y})$, you're not getting any quantitative result right? I mean, you get the regularity modulus, but given any arbitrary $x\notin F^{-1}(y)$, you can't really say anything about the distance $d(x,F^{-1}(y))$ because you don't know which neighbourhoods for which the regularity modulus holds. I tried looking around but there doesn't seem to be anything regarding the actual neighborhood for which the regularity modulus holds. I think the 'radius of metric regularity' refers to a different thing, regarding the choice of $F$, not the values in the neighborhood for which it holds. Am I missing something about the quantitative value of having a regularity modulus?
