I think it would ultimately depend on a use case for that metric, otherwise you're putting the cart before the horse. There are many measurements and calculations you could come up with, but no obvious (to me, anyway) interpretation of "most discontinuous": something is either in one piece or not. If you needed a metric like this for a practical purpose, your specific needs would be a starting point for designing one. If it's more of a shower thought, you sort of have "too much freedom" to be able to define anything that's necessarily meaningful.

Simple examples would be just "number of 'discrete parts'", "minimal area needed to span all territories" and things like that. Maybe you're more interested in "total distance from all satellites to wherever the capital is" or something, in a different context. The point is they'd all tell you radically different things, so it's important to know which one to ask for.

You could argue that something like Hawaii and Alaska's distance from the rest of the US makes the US score highly.
You could argue that any number of island nations score highly because after all, most of e.g. the US is in one part.
You could argue e.g. Norway's territories near both poles make it pretty high-scoring too.
You could argue that for whatever reason, distribution of area and population matter, and so on.

Maybe we look at the ratio of country perimeter to area? Counting the number of exclaves could also be a factor. And maybe a ratio of the distance to cover all the exclaves divided by their area?

So if a country were a perfect circle it's perimeter to area ratio would be 2/r, it has zero exclaves and then it's width would be the diameter.

If a country were two perfect circles of the same diameter, separated by a distance of the same diameter, it's area ratio would be 2/r, exclaves would be one, and it's width would be three times it's diameter.

So now you can imagine a country like Chile, modeled as a really skinny rectangle, has a pretty large perimeter to area ratio, no exclaves, and a width roughly the length of the rectangle.

I guess you'd have to decide if archipelago nations are measured as the geometry of the sea they own, or as discrete islands.

Step 1: Find the area of each chunk. The biggest chunk is your main chunk.

Step 2: Find the distance between the closest edge of main chunk and the center of each other chunk individually.

Step 3: Discontinuity of each chunk is area of chunk * distance from main chunk / total area.

Step 4: Total discontinuity is sum of each chunk's discontinuity.

Bolded parts are important. If you use the center of the main chunk, larger main chunk radii make other chunks seem more discontinuous than they should be. If you use the closest edge of other chunk's, you don't account for the entire area of the other chunk.

This will give you a number that is bigger when there are more and/or bigger pieces that are further away, and smaller when the opposite is true, normalized for the total area of the country so bigger countries aren't penalized just because they're bigger.