I have come across the following two statements.
By Riemann Roch theorem on $\mathbb CP^1$ due to g=0, we have dim $H^0(P)=2$ where P is seen as the divisor $1.P$, but $H^0(0)=1$, here 0 is the trivial zero divisor.
This means that there is a function with exactly one pole at P and no other poles.
Residue Theorem : Sum of residues = 0
So it seems like there can't be a function with exactly one pole at exactly one point.
What am I misunderstanding? Kindly clear my doubt.