One example in Michigan I can think of that continues as the same route number into the next state is M-49 and OH-49, a few miles east of the tri-point between Indiana, Ohio and Michigan. Michigan has a M-49 sign right away, Ohio's first OH-49 sign is about a half mile south of the state line and neither of them have an end sign.
M-99/OH-15 is the next one over and there is no end sign for either highway there either. Michigan simply has an M-99 sign and Ohio has a OH-15 sign again about a half mile from the state line.
At M-52 and OH-109 Ohio does a better job than Michigan at this one since Michigan doesn't have the Welcome to Michigan sign right at the state line and Ohio does. M-52's first sign is right next to the Welcome to Michigan sign. You can see the Michigan sign from the state line but it should be closer to the state line. The Michigan sign is a little north of the state line on US-24 too.