Sideband excitation near the carrier determines the minimum spectral width for steady-state free-electron laser oscillators fed by continuous electron beams. A sideband separated by deltaomega from the carrier resonates with harmonics of the upshifted bounce frequency for trapped particles, deltaomega = n2gamma(z)2OMEGA. The analysis focuses on sidebands excited in the immediate vicinity of the carrier deltaomega --> 0, in resonance with particles trapped near the separatrix, OMEGA --> 0. For electrons distributed uniformly around their orbits, the growth tends to zero as deltaomega, OMEGA --> 0, despite the infinite number of contributing harmonics. However, the distributions produced by injected electron beams are nonuniform around the trapped orbits, yielding finite growth rates GAMMA. Stability depends on the nonlinear shift deltak0(a0,omega0) of the carrier wave number from the empty cavity value, where the carrier amplitude a0 and frequency omega0 parametrize the free-electron laser (FEL) operation point. The curve deltak0(a0, omega0) = 0 divides the FEL parameter space into areas stable and unstable to sidebands. If deltak0 is negative, near-the-carrier sidebands are stable, and the linewidth is limited only by quantum effects. If deltak0 is positive an unstable frequency band can emerge around the carrier, of width DELTAomega congruent-to 8gamma(z)2upsilon0 deltak0, and maximum growth rate GAMMA(max)/k0 congruent-to (1/6)[2piN(deltak0/k0)]2, where N is the number of wiggler periods. The minimum linewidth is DELTAomega if the frequency separation between cavity modes is less than DELTAomega. ''Single mode'' operation in the unstable region is still possible if the cavity mode separation exceeds the unstable bandwidth DELTAomega. The above stability conclusions do not apply to sidebands ''far'' from the carrier DELTAomega approximately 2gamma(z)2OMEGA0. The latter poses less of a threat to FEL operation, since they are easier to filter out.
