Well yeah, they killed Palpatine, which meant the empire was done. Even the celebration at the end stated as such.

In the expanded universe the Empire survives as various warring states loosely united under a common goal. I guess a similar approach will be adopted in the next movie. After all, it is unlikely that the Empire will be defeated that easily. Severely crippled, yes, but not fully defeated.

I feel as though I should mention I've been reading Shadows of the Empire lately.

A pretty popular book. Xizor makes an interesting villain.

Albeit a short-lived one. This is actually something I've been meaning to get to reading for a long while, I just found it at one of my favorite used book shops for a song a couple of weeks ago and said "To entropy with it, let's do this."

Now if only I could find a copy of Heir to the Empire that wasn't falling apart or had a big chunk seemingly bitten out of it.

Oooh, neat. That's one of the few things I haven't gotten around to, but really probably should. Even if it no longer exists *sigh*.

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Also, I don't get how anyone could feasibly thing that just because Palpatine was killed that the Empire would suddenly just up and disappear. That isn't even remotely realistic. Disregard all expanded universe even. It just doesn't make sense for the rulers of nearly the entirety of the galaxy to just decide "yeah okay we'll leave power now and you'll never ever see anything from us again."

Well they also lost Vader and the crew of the Executor which would have been some of the most competent officers the Empire had as shown in Empire Strikes back since Vader kills officers for screwing up. Not to mention I doubt Palpatine had a successor in mind in case he died and even if he did then their woud be too much in fighting amongst the rest of the higher ups for them to do anything

Sorry, but I don't see the expanded universe as anything more than fanfiction. If it hasn't been stated in the movies or the canon cartoons, it isn't official.

I did not try to argue the Expanded Universe is canon, and besides it isn't anymore. I am merely stating that a similar explanation could be used and canonised for the new series of movies, as it would make sense.

Precisely this. Even discounting the entirety of the Expanded Universe, it would still be reasonable to expect the Empire to remain strong after Palpatine's death.