Separately holomorphic functions

Fix $m\ge 1$, let $U\subseteq {\mathbb{C}}^m$ be open and let $f:U\to \mathbb{C}$. For $a\in U$ and $k<m$ write

$U_{a,k}:=\{\zeta \in \mathbb{C}:(a_0,\dots ,a_{k-1},\zeta ,a_{k+1},\dots ,a_{m-1})\in U\}$

and let $f_{a,k}:U_{a,k}\to \mathbb{C}$ be the $k$th slice $f_{a,k}(\zeta )=f(a_0,\dots ,a_{k-1},\zeta ,a_{k+1},\dots ,a_{m-1})$.

The function $f$ is separately holomorphic on $U$ when for every $a\in U$ and every $k<m$ the slice $f_{a,k}$ is holomorphic on $U_{a,k}$ in the one-variable sense of Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions.

Each $U_{a,k}$ is open: if $\zeta \in U_{a,k}$, the corresponding point of $U$ has a ball $B(\cdot ,\rho )\subseteq U$ by The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, and changing only the $k$th coordinate by less than $\rho$ moves the point by less than $\rho$ in the norm of Complex $m$-space and its real coordinate dictionary, so the disc $\{\xi :\mathrm{\mid }\xi -\zeta \mathrm{\mid }<\rho \}$ lies in $U_{a,k}$ (Open ball, closed ball and sphere in a metric space).

No continuity in the remaining variables is asked. The condition constrains each slice separately and says nothing about how the slices fit together: a separately holomorphic function is not assumed continuous as a function on $U$, and on this page the two theorems that supply joint regularity — Osgood's lemma under continuity, and the locally bounded theorem — are what close that gap.

The slice through a point of a polydisc is a disc. If $U={\mathrm{\Delta }}_r(a^{\mathrm{\prime }})$ is a polydisc (Balls, polydiscs and the distinguished boundary in ${\mathbb{C}}^m$) and $a\in U$, then $U_{a,k}$ is the disc $\{\zeta :\mathrm{\mid }\zeta -a_k^{\mathrm{\prime }}\mathrm{\mid }<r_k\}$, which is what lets the one-variable theory be applied one coordinate at a time with the others held fixed.