Did French officials have any authority in occupied France?

I'm doing research and I need to get a feel for what role French officials
such as mayors, policemen, etc., played in German-occupied France in WWI.
I know that many people fled their homes, but some did not. For those that
did not, then, how did the day-to-day running of things work? Did French officials
retain some authority and work with the Germans in the area, or was the
military running everything? Specifically, if a murder took place in German territory, would there be any French police, detectives, etc., involved in solving it, or would it be, say, German military police instead?
Any help will be very appreciated! All the best,
Alex

Alex,
In occupied territory French police and authorities kept in function. German police was only for the German soldiers and military affairs. Vous comprenez. Je suis belge et c'était comme cela dans mon pays.
BD.

Vous vous souvenez, Bernard ?