I'd flip your comment about optical surfaces: they need to be smooth, not continuous, and smoothness is actually not required either! The requirement is a consistent phase relationship, which allows the signals to add together nicely via interference. Rays coming from same direction take the same path to the detector and interfere constructively, while rays coming from different directions tend to cancel.

For optical frequencies, the phase is difficult to measure directly, so we instead polish the surfaces down to a fraction of the wavelength of light (so that it all has the same phase). For radio telescope, the frequency is a lot lower, and we actually can measure the phase directly, so we can make our sensors crazy shapes and adjust it by adding delay. If you can change the individual delays (say, via software) you can change how they interfere and therefore change the sensitive direction for your telescope. This is how phased arrays function.