How to find out if a piecewise function is partially computable?
Page 3 of the book says: "partial function on a set S is a simply function whose domain is a subset of S."

The difference between $H_3(x)$ and $H_4(x)$ are the sets of $B$ and $C$ that you are going to make. for example if you give $B$ such that for all $x$ the other wise don't happen, then $H_3(x)$ will be a total function and we know every total functions are partial functions since $S \subseteq S$. 

between $f_1(x)$ and $f_2(x)$ the one is partial computable which gives output for some $x$s in its domain.