Composition of partial functions

For partial functions $h:X\rightharpoonup Y$ and $g:Y\rightharpoonup Z$, their function composition is defined precisely onFor example, composing $h(x)=1/x$ with $g(y)=\sqrt{y}$, as real partial functions, gives $1/\sqrt{x}$ only on $x>0$.

More generally, strict composition of partial functions makes $g(h_1(x),\dots ,h_r(x))$ defined only if every $h_i(x)$ is defined and $g$ is defined on the resulting tuple. An argument discarded by the outer function must still be defined: the expression $g(1/x)$ with the everywhere-zero constant function $g$ remains undefined at zero. A representation in a programming language using lazy evaluation therefore needs an explicit check of the intermediate definedness conditions.