Is a total function also a partial function?

Solution 1:

When someone says "partial function", the usual interpretation is that the function may or may not be defined on the entire domain. The word is also sometimes used with the meaning "not total", but that meaning is relatively rare and will usually only be understood in contexts where the ordinary meaning would be clearly senseless.

The unambiguous way to say that a function is not total is "not total".

Note that in almost all mathematical subfields, the word "function" alone means "total function"; we only add the word "total" when there's a risk that the reader might otherwise think we were allowing non-total ones, too.

Solution 2:

Yes. A partial function from $A$ to $B$ is a total function $U \to B$ for some $U \subseteq A$; under this definition a total function is just a partial function with $U=A$. This is valid since $A \subseteq A$.