We have just started quantum chemistry/mechanics, so I am not familiar with eigenvalues or Hamiltonian. However, the question was given exactly as I wrote, with no mention of permitted values or ranges.

If I am understanding this right, if a function is finite or square integrable, it simply means that is does not approach infinity for all values of x correct? No matter what value of x the function takes on, it will have a single, non-infinity value?