Extending the classical result that the roots of a polynomial with coefficients in $\Bbb C$ are continuous functions of the coefficients of the polynomial, nonstandard analysis is used to prove that if $\mathcal F = \{f_\lambda : \lambda\in ∈ \Lambda\}$ is a set of polynomials in $\Bbb C[t]$ and if ${}^∗\mathcal G = \{g_\lambda : \lambda\in\Lambda\}$ is a set of polynomials in ${}^∗\Bbb C_0[t]$ such that $g_\lambda$ is an infinitesimal deformation of $f_\lambda$ for all $\lambda\in\Lambda$, then the nonstandard affine variety ${}^∗V_0(\mathcal G)$ is an infinitesimal deformation of the affine variety $V(\mathcal F)$.
Mel Nathanson