First of all, rigorously speaking, as the spectrum of the position operator $X$ on $L^2(R)$ is purely continuous, the spectral measure $P_E$ is labeled by Borel sets $E\subset R$, so, in particular $E$ can be an interval $[a,b]$.
In position representation:
$$\left(P_E \psi\right)(x) = \chi_E(x) \psi(x)\quad \forall x \in R$$
where $\chi_E(z)=1$ if $z\in E$, or $\chi_E(z)=0$ otherwise.
When measuring the position of the particle, if the precision of the instrument is $2\delta>0$, so that one cannot distinguish anything inside the interval $[x_0-\delta, x_0+\delta]$, the wavefunction immediately after the measurement is, up to normalization,
provided $\psi$ was the wavefuntion before the measurement and the found position was $x_0$ (taking the precision of the instrument into account).
This is nothing but a particular case of the von Neumann - Luders axiom on quantum measurement that includes both observables with point spectrum and continuous spectrum. In the second case the notion of eigenvector cannot apply and, however, it is by no means necessary. Just the notion of spectral measure associated with a self-adjoint operator is sufficient.
The fact that real instruments for observables with continuous spectrum are truly described by that axiom, even taking the precision into account as done before, is however questionable due to many practical reasons.
It is more plausible that, in real (non-destructive) experiments of position the wavefunction after the measurement is obtained from the incoming one through a so called quantum operation (http://en.wikipedia.org/wiki/Quantum_operation).