them to our family $f_s$, as follows: $G_s$ is the general fiber of $f_s$, $F_{s,t}:= f_s^{-1}(t)$, $\mu(s)$ is the total Milnor number and $\lambda(s)$ is the total Milnor-Lê number at infinity of $f_s$. Note that if $f_s$ is GI then $\lambda(s) =0$ and $\chi(F_{s,t}) = \chi(n,d)$ if $t\not\in B_{f_s}$. We first have the following semi-continuity result:

\[l:semicont\] Let $P$ be a [[$\mathcal F$[[ISI]{}]{}]{}]{} deformation of $f_0$. Then the number of vanishing cycles $\gamma(s) := \mu(s) +\lambda(s)$ is upper semi-continuous, i.e. $$\mu(s) +\lambda(s) \ge \mu(0) +\lambda(0).$$

When examining the formula (\[eq:chi\]) applied to $f_s$, namely: $$\label{eq:gammas}
   \gamma(s) = b_{n-1}(G_s) = (d-1)^n - \sum_{p_i\in W_s}(\mu_{p_i, gen}(s) + \mu_{p_i}^{{\infty}}(s)),$$ we see that $\mu_{p_i, gen}(s)$ and $\mu_{p_i}^{{\infty}}(s)$ are lower semi-continuous since local Milnor numbers of hypersurface singularities.

From (\[eq:gammas\]) and the semi-continuity of its $\mu$-ingredients, we also get:

\[c:semi\] If $P$ is a $\gamma(s)$-constant deformation, then both $\mu_{p_i, gen}(s)$ and $\mu_{p_i}^{{\infty}}(s)$ are constant for small $|s|$. [${_\square}$]{}

Let us denote by $\mu_I(s)$, $\mu_{II}(s)$, $\mu_{III}(s)$ the sum of the Milnor numbers of the singularities of $f_s$ at the affine points of $\Gamma_I$, $\Gamma_{II}$, $\Gamma_{III}$ respectively. We have $\mu(s) = \mu_I(s) + \mu_{II}(s) + \mu_{III}(s)$ and, if the deformation is general-at-infinity, then we know that $\mu(s) 