Domain of a partial function

For a partial function $f:X\rightharpoonup Y$, its domain of a partial function is the subsetIts restriction to $D$ is an ordinary function $D\to Y$. Outside $D$, function evaluation has no assigned value.

For instance, the real square root as a partial function on the real line has $D=[0,\infty )$. Both $0$ and $-1$ lie in the surrounding source, but only at zero is there an output; the value zero and absence of a value are different situations.