  We present the first demonstration of an inertially sensitive atomic
interferometer based on a continuous, rather than pulsed, atomic beam at
sub-Doppler temperatures in three dimensions. We demonstrate 30\% fringe
contrast in continuous, inertially sensitive interference fringes at
interrogation time $T=6.7~\mathrm{ms}$ and a short-term phase measurement noise
of $530~\mathrm{\mu rad /\sqrt{Hz}}$ as inferred from interference
measurements. Atoms are delivered to the interferometer by a cold-rubidium
source that produces a high flux of atoms at temperature $\leq15~\mathrm{\mu}
K$ in three dimensions while reducing near-resonance fluorescence in the
downstream path of the atoms. We describe the optimization of the interrogating
Raman beams to achieve high contrast, and validate interferometer operation
through comparison with measurements by commercial accelerometers. We further
provide a demonstration of zero-dead-time phase-shear readout of atom
interferometer phase, achieving a measurement rate up to 160~Hz. This
demonstration lays the groundwork for future gyroscope/accelerometer sensors
that measure continuously, with both high bandwidth and high sensitivity, and
on dynamic platforms.
