Another point to be careful about is 0 Hz is DC. No function generator reliably goes that low. 0 Hz would be a battery or just a connection to ground. Consider what actual AC frequency your lowest need would be: 1 Hz, 0.1 Hz, 0.01 Hz, etc. And keep in mind that the lower you go, the more expensive it will be.
On the other hand, almost any function generator will go up to the top end of the audio range or 20,000 Hz so you should have no problems there.
A popular range for an inexpensive generator may be from 20 Hz to 100 KHz. I have an inexpensive one that has a range of 20 Hz to 150 KHz.
But here's a thought for you. In today's world, many DSOs (Digital Storage Scopes) come with a waveform generator. I have a two channel, 150 MHz DSO that has such a generator. It has about 5 different programmed waveforms, DC, and four arbitrary, programmable ones. The frequency is adjustable from 0.01 Hz to 5 or 10 MHz with coarse and fine controls. Voltage level up to around 8 Volts, P-P. And it only cost around $250 US. The only down side is the waveforms are digitally generated, probably with an 8 bit D-A so if the accompanying digital noise is not something you can work with, then it isn't for you. Personally I find it very useful and it's great having both scope and waveform generator in one unit. At the $250 price level it is considered an entry level scope but it can do many things that the Tektronix and HP scopes I have worked with, that cost many thousands of dollars, simply could not do. Here's a link if you want to see it.
https://www.amazon.com/dp/B08Y6KVMSV?tag=bgus-267707-20&th=1Rigol and Siglent also have lines of low end scopes but I can not suggest any model numbers.
I don't know about calibration, but I don't see why not.