The authority of the officer arises out of state law. It sounds like the officer's observations were in his jurisdiction anyhow and so the alleged "offense" occurred within the officer's jurisdiction. When things get dicey and there is a jurisdictional challenge over a county or municipal line, what the police can and do do in some circumstances is to call law enforcement in the second jurisdiction and they actually file the charges, using the officer from the first jurisdiction for the purpose of testimony. I don't think this is required in your case.
The old fasioned crossing the county line to get away from the law made good movies.
This answer is offered for informational purposes only. It is not offered as, and does not constitute, legal advice. Laws vary widely from state to state. You should rely only on the advice given to you during a personal consultation by a local attorney who is thoroughly familiar with state laws and the area of practice in which your concern lies.
Michigan law provides that an officer may follow and a violator he has witnessed commit an offense, even if the stop occurs outside of the geographical limits of the officer's jurisdicition.
Without knowing more about your case, I can't really comment on whether you have somewhat of a leg to stand on, but this statute would appear to address your concern.