Position representation

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

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: