  We study how Betti numbers of ideals in a local ring change under small
perturbations. Given $p\in\mathbb N$ and given an ideal $I$ of a Noetherian
local ring $(R,\mathfrak m)$, our main result states that there exists $N>0$
such that if $J$ is an ideal with $I\equiv J\bmod \mathfrak m^N$ and with the
same Hilbert function as $I$, then the Betti numbers $\beta_i^R(R/I)$ and
$\beta_i^R(R/J)$ coincide for $0\le i\le p$. Moreover, we present several cases
in which an ideal $J$ such that $I \equiv J \bmod \mathfrak m^N$ is forced to
have the same Hilbert function as $I$, and therefore the same Betti numbers.
