I noticed today after I updated the firmware on my P4P that my video clip naming got reset back to 0001. Is this always the case after an update? How can I keep it continuous. It was set to continuous in the DJI Go app but somehow got reset. And can I reset it to pick up where it left off?

You should always check all your settings after a firmware update as many of them can be reset. The latest firmware update had fixes for the camera so all the settings were reset.

Go into your camera settings and under the cog icon, you'll find an ient labelled File Index Mode.
Set it to Continuous

Labroides is spot on. Please change it as Continuous in camera settings.

I also find it quite annoying that the standard setting is Reset, i keep forgtting to check that, DJI please consider a change in this behavior.
It can happen at FW upgrades and will happen when you reset the camera settings and/or AC.

I have found that you can get it back on count, i do it like this:
Keep a few files (i keep one .mp4, one .JPG and on .DNG) rename to the number it should have had (same no for .DNG and .JPG, +/- 1 for .MP4).
Start up RC and AC, change setting to continuous, shut down AC, insert SD with renamed files and start up AC.
Take a picture and it should have the next number in you preferred count.

Not sure you need to start up AC without SD when doing the change, that is just how i know it would work.