Usually in QM we say that a wavefunction psi is an eigenfunction of some operator if that operator acting on psi gives eigenvalue * psi. The position operator is just "multiply by x". So any psi would seem to fit the above description of an eigenfunction of the position operator with eigenvalue x. But obviously this doesn't make sense because in general wavefunctions do not have definite positions. So where does the above reasoning break down? Is it to do with x being a continuous variable?