Most analog (really, with a geared mechanism) clocks do not "snap" on exact minutes but slowly drive toward them (because that's simpler and thus cheaper).

Analog Clock movements with second hands! The seconds hand is rarely smooth, we want the tick, but the minute hand and hour hands are smooth.

Except those of us (like my SO) who are bothered by the ticking sound at the edge of audibility, and prefer smooth seconds motion.

For those who remember Swatch watches, the movement was especially loud.

Which lead to a memorable uni moment where a female friend looked at me in frustration and said "if you don't like it, take it off".

Unless it's an old AC-motor clock

All escapement driven clocks are discrete.

This is true unless you look at something like a Seiko Spring Drive, which has a completely smooth second hand sweep, although it’s not entirely mechanical (and maybe only watch nerds care about this)

https://www.hodinkee.com/articles/does-spring-drive-have-an-...

Pedantically, only above second granularities. They're continuous between second hand sweep movements at subsecond ones, no? I mean, there's no point on the watch that the second hand doesn't "hit" at some point, however small.

Or am I wrong that "intermittent, jerky, continuity is still continuity"?

Well it depends on the interval. Taken to the limit, this would be a very nerdy way to rediscover calculus.