I checked fuse #38 which is the correct fuse for the inverter and it was okay. How I checked for continuity on the outlet is that I took the plug out of the dashboard trim and took the outlet apart, then using a multimeter I tested for continuity. I didn't get continuity from touching the posts on the outlet.
OK, just to be certain we're both on the same page.
You double checked your multimeter settings by shorting the test probes together, and then checked for continuity from each of the 3 AC terminals to the connector pins on the bottom of the assembly?
You could be an Engineer or an electrician but we don't know that and have to ask for the details.
Given that most hair dryers are about 1500 Watts or 10 x the rated output of the inverter, something may have been fried on that outlet assembly board before any current limiting was engaged.
When I next get to my PC, I'll see if I can find any info about that assembly. If it were me, I would try to inspect the circuit board for any fusing or fried traces, etc.
Could you post another close-up of the circuit board on the outlet assembly?
See if the circuit board can be removed from the plastic housing so you can see both sides.
You should be able to follow the path of each of the AC terminals through the PCB to the connector with your ohm meter and identify exactly where the open exists.
I can likely teach how to repair the circuit board if the board is fried but it does depend on the damage.
After another magnified look, it's difficult to see the circuit board using my phone but it does appear to have micro circuitry including semi-conductors. This type of assembly can often be impossible to repair.
I would be very careful and try to test for voltage at the connector feeding the outlet assembly.
Measure AC voltage between a good ground and the female side of the connector using an insulated paper clip or whatever else that fits better.
Use alligator test clips to hold the paperclip and apply electrical tape to insulate. And don't rest your elbow on a metal part while you're holding the taped paperclip.
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