 Solution

Away from $p,q,r$, all three coordinate functions are holomorphic and the first is one, so they define a projective point. Near $p$, take a local coordinate $z$ vanishing there. Write $f=a/z+O(1)$ with $a\ne 0$, while $g$ is holomorphic. ThenThe scaled coordinates are holomorphic and do not all vanish, proving a holomorphic extension. Similarly $F(r)=[0:0:1]$. At $q$, write $f=b/z+O(1)$ and $g=c/z+O(1)$ with $b,c\ne 0$. The same scaling gives $F(q)=[0:b:c]$, again well-defined. Hence