The terminology is a bit confusing, so let me try and clarify. From your pictures, you seem to be looking at a chirp sequence radar. In such a signaling scheme, a sequence of linear FM signals (chirps) are transmitted. Each chirp in itself can be considered a "FMCW measurement", yielding a signal with a certain frequency (corresponding to the object distance and a Doppler-term) and phase. In order to resolve velocities of objects, the phase difference between the chirps in the sequence can be evaluated. I will not derive the whole thing here, you can read up on the basics yourself, e.g. here (module 1 is about FMCW, modules 2&3 explain the essence of the chirp sequence approach for velocity measurement).
Now to answer your question, yes you can have a "gap" between subsequent transmit chirps of a sequence. In fact, there probably should be, so as to have some time for the signal generator to "ramp down" before it can "ramp up" again. How much this gap should be depends on your application. The maximum unambiguous velocity you can measure is given by:
$v_{\mathrm{max}} = \frac{\mathrm{c}}{4f_0T_{\mathrm{p}}}$ ,
where $\mathrm{c}$ is the speed of light, $f_0$ the carrier frequency of your transmit signal, and $T_{\mathrm{p}}$ the chirp repetition rate (duration of chirp + what you define as the gap). The velocity resolution is then accordingly $v_{\mathrm{max}}$ divided by the number of chirps you transmit in a sequence.
Of course this is not an exhaustive answer, but hope it helps.
P.S. @Jason R: more precisely, I believe continuous refers to the transmit signal being of a duration much longer than the delay from object reflections so that transmission and reception happens at the same time (as opposed to pulsed radar where you essentially send a very short pulse and then "wait" for the reflections).