Gauss–Kronecker curvature and equisingularity at infinity of definable families

Assume given a polynomially bounded o-minimal structure expanding the real numbers. Let $(T_s)_{s\in \mathbb{R}}$ be globally definable one parameter family of $C^2$-hypersurfaces $\mathbb{R}^n$. Upon defining notion generalized critical value for such we show that functions $s \to |K(s)|$ and $s\to K(s)$, respectively total absolute Gauss-Kronecker curvature $T_s$, are continuous in any neighbourhood which is not critical. In particular this provides necessary criterion equisingularity levels polynomial.