Abstract: Given a (real or complex) analytic map $f=(f_1,\dots,f_n):X\to\mathbb{K}^n$, one can consider its approximations by maps $T_d(f)$ whose coordinates are Taylor polynomials of the $f_i$ of degree $d$. We will show that the continuity of the family of fibres of $f$ is finitely determined. That is, it is already determined by the polynomial maps $T_d(f)$ for $d$ sufficiently large. As a consequence, we also obtain finite determinacy of analytic complete intersections. This is joint work with H. Seyedinejad.