[proofplan] We prove that holomorphic functions of $n \geq 2$ complex variables have no isolated singularities: if $f \in \mathcal{O}(\Omega \setminus \{p\})$ for a point $p \in \Omega$, then $f$ extends to a [holomorphic function](/page/Holomorphic%20Function) on all of $\Omega$. This is a direct application of the [Removability of Codimension-Two Singularities](/theorems/3383) theorem, after verifying that a single point has the required codimension. [/proofplan] [step:Verify that $\{p\}$ is a compact analytic subset of complex codimension $\geq 2$] The singleton $A = \{p\}$ is a compact subset of $\Omega$. It is an analytic subset of $\Omega$: it is the common zero set of the $n$ holomorphic functions \begin{align*} h_j: \Omega &\to \mathbb{C} \\ z &\mapsto z_j - p_j, \quad j = 1, \ldots, n. \end{align*} The set $A = \{z \in \Omega : h_1(z) = \cdots = h_n(z) = 0\} = \{p\}$ has complex dimension $0$ (it is a single point). Since $\Omega \subset \mathbb{C}^n$ has complex dimension $n$, the complex codimension of $A$ is $n - 0 = n \geq 2$ (using the hypothesis $n \geq 2$). [guided] Why is the codimension condition $n \geq 2$ needed here? A single point in $\mathbb{C}^n$ has complex dimension $0$, hence complex codimension $n$. When $n = 1$, this codimension is $1$, which does not meet the threshold $\geq 2$ required by the removability theorem. Indeed, in one complex variable, isolated singularities are not automatically removable: the function $z \mapsto 1/z$ is holomorphic on $\mathbb{C} \setminus \{0\}$ but does not extend to $\mathbb{C}$. The theorem is genuinely a phenomenon of several complex variables ($n \geq 2$). The set $\{p\}$ is an analytic set because it is cut out by holomorphic equations. Explicitly, $\{p\} = \{z \in \Omega : z_1 = p_1, \ldots, z_n = p_n\}$, the zero locus of $n$ independent linear holomorphic functions. These $n$ equations impose $n$ independent conditions on $n$ complex variables, leaving a $0$-dimensional solution set. [/guided] [/step] [step:Apply the removability theorem to extend $f$ across $\{p\}$] Since $A = \{p\}$ is a compact analytic subset of $\Omega$ with complex codimension $n \geq 2$, and $f \in \mathcal{O}(\Omega \setminus A)$, the hypotheses of the [Removability of Codimension-Two Singularities](/theorems/3383) theorem are satisfied. That theorem provides a unique [holomorphic function](/page/Holomorphic%20Function) $\tilde{f} \in \mathcal{O}(\Omega)$ with $\tilde{f} = f$ on $\Omega \setminus \{p\}$. [guided] This result should be compared with the Riemann removable singularity theorem in one variable, which requires the additional hypothesis that $f$ is bounded near the singularity. In several variables ($n \geq 2$), no boundedness assumption is needed: every isolated singularity is automatically removable, regardless of the growth of $f$ near $p$. The underlying reason is topological. In one variable, removing a point from a disc creates a punctured disc, which has nontrivial topology (its fundamental group is $\mathbb{Z}$). A function like $z \mapsto 1/z$ exploits this topology to have a pole. In several variables with $n \geq 2$, removing a point from a ball creates a set that is still simply connected (the sphere $S^{2n-1}$ with $n \geq 2$ is simply connected). The Hartogs extension phenomenon forces holomorphic functions on punctured domains to extend automatically. Uniqueness of the extension follows from the [Identity Principle](/theorems/3357): any two extensions agree on the connected [open set](/page/Open%20Set) $\Omega \setminus \{p\}$, hence on all of $\Omega$. [/guided] [/step]