If dissonance and noise, or more precisely, the emancipation of dissonance and noise formed the contested ground of music in the first half of the 20th century, continuity was the central issue of the second half and we're still wrestling with it.
After the very brief moment of radical serial technique (part of that line that goes from Webern and Messiaen to the early numbered works of Goeyvaerts, Boulez of the first book of Structures etc. through the first (gorgeous failures that they were) attempts at synthesis with superimposed sine waves) in which discrete events were the focus, several competing approaches emerged for "connecting the dots."
The place of John Cage in this chronology is important (and it's also important to insist that Cage's music was, in part, parcel of the serial program with his charts equivalent in many ways to 12-tone arrays, including, in some cases, insistence on full 12-tone aggregates (although said aggregates may not have survived from the chart to the final surface of the composition (but then again even Milton Babbitt had his "weighted aggregates"...))) and it was Cage's unique insistence on the importance of silence to the problem of continuity. Cage came to silence in his compositional practice via musical structure, with his formal schemes, from the early square root pieces (the measure length structure of which is announced on many scores in a fashion precisely analogous to the announcement of tonality in a classical tonal score) to the later works in which clock time replaces metric proportions, a movement towards ametricality akin to that towards atonality. These schemes were usually pre-compositional, the blank, or silent form into which sounded events were placed or left empty. The musical potential of that emptiness became an increasingly attractive field for the composer, evidenced most vividly in the progression towards greater durations of silence over the three movements of the Concerto for Prepared Piano and Chamber Orchestra and, of course, 4'33" which was actually composed out of formal methods associated with Music of Changes.
In addition to Cage, as a young composer, it was the music and ideas about continuity of Christian Wolff and La Monte Young. Christian's article On Form in die reihe, radically opened up continuity through plain assertion (most plainly with the idea that a piece could be constrained simply by length of available time on a concert program), much as Stravinsky —in his Satie/Debussy tradition — asserted a tonality. And La Monte, with his idea of "getting inside a sound", was almost Christian's opposite, closing continuity to the acoustically intimate, by allowing (framing the articulation of the music) the ear to attend to ever more detail within a sustained sound event. There are, of course, many innovations by others in both these lines — the acoustical graffiti and resultant patterns emerging from early Reich and Glass, Johnston and Tenney's proportionate explorations, Lucier's beats and Music on a Long Thin Wire, etc. all represent extensions and refinements of the "inside a sound" line, for example, while the game-like strategies of a John Zorn, for example, extend the Wolff model.
With my own music, I readily plead inconsistency, but bouncing materially between the poles of the radically continuous and discontinuous seems to still be productive and musically rich, while at the same time, having a certain degree of agnosticism towards continuity, I find myself frequently using metric- or clock time-based formal systems, like those of Cage and Lou Harrison.