I tested this out on my own SQL server, and you're right - a normal column will return table information, but using a MySQL function will not. I guess that's because when you're using a function, you're not actually talking about data from the table - you're actually SELECT'ing a computed value based on the outcome of a function.
So really, there isn't a table associated with the data from an AVG/MAX/etc. function - it's just data that has been calculated.