30A continuous on a 40A breaker, what gauge wire do I need?

You should generally not use anything smaller than #8 AWG on a building circuit protected by a 40A breaker, regardless of what the actual load is. If you're talking about automotive loads, different numbers would apply and you could use 10 AWG or even 12 AWG for short runs (under 7 ft).

40 amp circuit needs 8 gauge wire.

FYI, using the 80% rule, the most current that can be continuously supplied by a 40 amp circuit is 32 amps, so you just made it.

The wire size is related to the circuit breaker value, not the load. If it was a long run, I'd use 6 gauge.

I'd use ten gauge stranded. Always use stranded over solid. You could get away with 12 gauge, but for a really good answer I'd have to know what the expected load is going to be. 12 stranded will handle 30A if that's what you're saying, however, is that the max load? Even at startup?

for a 40 amp breaker use #8 wire

http://www.cerrowire.com/ampacity-charts

8AWG.