Let $\Omega$ be a domain in $\mathbb C$, and let $\mathscr X$ be some class of functions from $\Omega$ to $\mathbb C$. A set $E\subset \Omega$ is called removable for holomorphic functions of class $\mathscr X$ if the following holds: every function $f\in\mathscr X$ that is holomorphic on $\Omega\setminus E$ is actually holomorphic on $\Omega$, possibly, after being redefined on $E$.
(An example of the above: $E$ is a line interval, $\mathscr X$ consists of continuous functions. In this case $E$ is removable, which is shown in the answer.)
It is clear that the larger $\mathscr X$ is, the smaller is the class of removable sets. In the extreme case, if $\mathscr X$ contains all functions $\Omega\to\mathbb C$, there are no nonempty removable sets. Indeed, if $a\in E$, then $f(z)=\frac{1}{z-a}$ (arbitrarily defined at $z=a$) is holomorphic on $\Omega\setminus E$ but has no holomorphic extension to $\Omega$.
The problem of describing removable sets is nontrivial in many classes $\mathscr X$ such as
- $L^{\infty}(\Omega)$, bounded functions
- $C(\Omega)$, continuous functions
- $C^{\alpha}(\Omega)$, Hölder continuous functions
- $\mathrm{Lip}(\Omega)$, Lipschitz functions
Which sets are removable for holomorphic functions in these classes?