The initial-to-final amplitudes are ill-defined if one thinks of the continuum limit right from the beginning, because the fluctuations in the field can become unbounded. So the path-integral can be thought of as on a discrete square lattice, with lattice spacing a and the limit a → 0 should be taken carefully. If the final results do not depend on the shape of the lattice or the value of a, then the continuum limit exists.