The continuum limit of the lattice fermion version of the anisotropic spin-1/2 Heisenberg chain is reconsidered. It is shown that certain matrix elements of the Heisenberg Hamiltonian converge towards corresponding matrix elements of a massive Thirring model as previously suggested by Luther and Peschel. However, the result is only to first order consistent with the exactly known spectral and critical properties of the two models. Going beyond previous results in addition to the coupling constant of the massive Thirring model the kinetic energy coefficient comes out correctly to the first order of the lattice fermion interaction, too. Emphasizing the role of the underlying Hilbert spaces the discrepancy in higher orders is explained.