On a family of mild functions

Abstract: We prove that the function $P_\alpha(x) = \exp(1-x^{-\alpha})$ with $\alpha > 0$, is $1/\alpha$-mild. We apply this result to obtain a uniform $1/\alpha$-mild parametrization of the family of curves ${xy = \epsilon^2 \mid (x,y) \in (0,1)^2}$ for $\epsilon \in (0,1)$, which does not have a uniform $0$-mild parametrization by work of Yomdin. More generally we can parametrize families of power-subanalytic curves. This improves a result of Benjamini and Novikov that gives a $2$-mild parametrization.