Alright, so I've always thought that a partial function was a function from $A$ to $B$ whose domain is only a subset of $A$. A total function, on the other hand, I took to be a function whose domain would be the whole of $A$ in the aforementioned case. So these two properties (i.e. partial vs. total) seemed mutually exclusive to me.
A recursive function, or so I supposed, is a function which is defined by a recursion equation (i.e. whose values are defined by previously defined values, etc).
Naturally then, I always thought that a partial recursive function was just a partial function that was defined by a recursion equation (i.e. previously defined values, etc.)
But now I'm being told that there are totally defined partial recursive functions?!? I tried to find some information on it, but all I found were a bunch of authors who just annoyingly assert this fact without further explanation. Example: "There are, of course, partial recursive functions that are totally defined..."
It seems like I'm missing something really simple here, someone please explain what I'm getting wrong.