Can a voltage source be DC and generate a sawtooth signal?
Just don't get too hung-up on calling something ac or dc. If there is an ac signal superimposed on a DC level then that is exactly what it is, an ac signal + a dc level.

Calling that composite waveform either dc or ac is missing the point.

You wouldn't call a battery an ac source even though it gradually discharges and then possibly is recharged - that would be what could be described as a signal with dc and a sawtooth.

Give it a full description is my take on things.