We discuss the phase transitions of fermions in one dimension with a narrow Feshbach resonance described by the boson–fermion resonance model. By means of the bosonization technique, we derive a low-energy Hamiltonian of the system and show that a strongly correlated state exists, where the order parameters of the Bose condensation and superfluidity decay with the same critical exponent. We also show that density fluctuations near the Fermi wavevector are strongly suppressed as a consequence of a spin gap and a gap against the formation of phase slips. We find a Luther–Emery point where the phase slips and the spin excitations can be described in terms of pseudofermions, providing closed form expressions of the density–density correlations and the spectral functions. The relevance of our results for experiments with ultracold atomic gases subject to one-dimensional confinement is also discussed.