 Solution

The reciprocal map is holomorphic for $z\ne 0$, and the complex exponential is entire. Its denominator vanishes exactly whenAll these points are excluded from $U$. The set $U$ is open: it is the inverse image of $\mathbb{C}\setminus \mathbb{Z}$ under the reciprocal map on $\mathbb{C}\setminus \{0\}$. A reciprocal of a nonvanishing holomorphic function is holomorphic. Therefore $f$ is holomorphic throughout $U$.