Function generator as a DC power supply
No, that won't work. 

You could use fixed voltage supplies (AC adapters, wall warts) for most power supply purposes - just understand that your circuits are living dangerously if the supplies aren't current limited, so be careful and double-check your connections - until you can afford an adjustable PSU. 

One option is to make one of your first projects an adjustable PSU with current limiting (using an LM317 and a heatsink) powered by an AC adaptor, perhaps a 19V one from an old laptop. It'll teach you about proper heatsinking!

So I'd still go for the function generator as a first purchase (assuming you already have a multimeter!) but use other arrangements to supply power.