A way to write the wavefunction $ψ(x)$ such that the position operator is:
i.e., a function that takes the wavefunction as input, and outputs another function:

$x$

$xψ(x)$

If you believe that mathematicians took care of continuous spectrum for us and that everything just works, the most concrete and direct thing that this representation tells us is that:

the probability of finding a particle between $x_{0}$ and $x_{1}$ at time $t$equals:

$∫_{x_{0}}xψx,tdx$