Holomorphic extension across a finite set of bounded singularities, preserving the bound

Statement. Let $U\subseteq\mathbb C$ be open, let $P$ be a finite set of complex numbers, let $H:\mathbb C\to\mathbb C$ be holomorphic on $U\setminus P$, and suppose $\|H(s)\|\le M$ for every $s\in U\setminus P$. Then there is a function $G:\mathbb C\to\mathbb C$ which is holomorphic on all of $U$, agrees with $H$ on $U\setminus P$, and satisfies $\|G(s)\|\le M$ for every $s\in U$ — including the points of $P$ that lie in $U$.

Discussion. This is Riemann's removable singularity theorem, iterated over a finite set and with the uniform bound carried across the removed points. The first two conclusions follow by induction on $P$: if $p$ is one of the points, then $U\setminus\{p\}$ is again open, the inductive hypothesis produces a holomorphic extension across the remaining points, and Riemann's theorem (in the form: a function holomorphic and bounded on a punctured neighbourhood of $p$ extends holomorphically across $p$) removes the singularity at $p$.

The third conclusion is not automatic from the first two and is what makes the lemma usable in contour-integration estimates: at a point $p\in U\cap P$ the value $G(p)$ is a limit of values $G(z)=H(z)$ with $z\in U\setminus P$, because $U$ is open and $P$ is finite, so $U\setminus P$ accumulates at $p$; continuity of $G$ at $p$ then gives $\|G(p)\|\le M$ as well.

Note that no hypothesis $0\le M$ is needed. If $U\setminus P$ were empty then $U$ would be a finite open subset of $\mathbb C$, hence empty, and the conclusion is vacuous.

Use. In the Hardy–Littlewood/Perron contour method one is given a generating function $H$ with its polar parts subtracted off, holomorphic and bounded by $M$ away from a finite set $P$ of would-be poles. This lemma says the shifted contour picks up no residues at all and that the bound $M$ is valid on the whole rectangle, which is exactly what the edge estimates require.