Let $r$ and $s$ be two strictly positive natural integers such that $r \geqslant s$. For $y \in ] 0,1 [$, we set
$$f _ { r , s } ( y ) = \int _ { 0 } ^ { 1 } \frac { x ^ { r } y ^ { s } } { 1 - x y } \mathrm {~d} x$$
Show that $f _ { r , s }$ is continuous and integrable on $] 0,1 [$.