Should the transmitted signal be continuous in FMCW?
Let us say the sweep repetition interval is $T_s$, it is the part with frequency modulation. If interval $T_i = T_s/N$ for large N, then drawbacks are: loss of average power, decrease of maximum measurable unambiguous Doppler frequency and it might cause amplifiers to introduce unwanted transients in the signal. Therefore, there is no advantage of adding an interval. 

However, if interval $T_i = N*T_s$ for large integer value of N, then it is no longer called an FMCW, rather it will be called LFM, linear FM modulated pulse signal. The processing of which is more complicated than that of FMCW.