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  "body": "\n## Statement\n\nLet $D \\subseteq \\C^n$ be a domain, that is, a nonempty connected open set.\nLet $f$ and $g$ be [[holomorphic functions|507C34W0ZT]] on $D$.\nIf $f = g$ on a nonempty open subset of $D$, then $f = g$ on all of $D$.\n\n## Remarks\n\nPut $h = f - g$ and let $E$ be the set of points of $D$ near which $h$ vanishes identically.\nThe set $E$ is open by definition and nonempty by hypothesis.\nIt is also closed in $D$.\nAt a limit point $a \\in D$ of $E$, all partial derivatives of $h$ vanish by continuity, so the power series of $h$ at $a$ is zero and $h = 0$ near $a$.\nSince $D$ is connected, $E = D$.\nThe one-variable condition \"agreement on a set with a limit point\" is not enough when $n \\ge 2$: $f(z_1, z_2) = z_1$ vanishes on the whole line $\\set{z_1 = 0}$ but is not zero.\nConnectedness is needed.\nLet $D = \\set{\\abs{z_1} \u003c 1} \\cup \\set{\\abs{z_1 - 3} \u003c 1}$ in $\\C$, or its product with $\\C^{n-1}$.\nThe function equal to $0$ on the first piece and $1$ on the second is holomorphic and vanishes on an open set without being zero.\n\n## Explanations\n\n### L1 human\n\nA holomorphic function on a connected domain is fixed everywhere by its values on any small ball.\nSo two such functions that agree on a small ball agree everywhere.\nThis is the rigidity that separates holomorphic functions from smooth real functions.\n\n### L2 human\n\nThis is the several-variable form of the [[identity theorem|0YBVKMYX0W]].\nThe proof uses only the local power series expansion and connectedness, exactly as in one variable.\nWhat changes is the size of the set needed.\nZero sets of holomorphic functions in $\\C^n$, $n \\ge 2$, are never isolated points, so agreement on a discrete set or even on a complex hypersurface says little.\nIt is enough that $f$ and $g$ agree on a set that is not contained in any proper analytic subset of $D$.\nAn example of such a set is a nonempty open subset of $\\R^n \\cap D$, when $D$ meets $\\R^n$.\nThe theorem underlies uniqueness of analytic continuation in several variables.\n",
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  "name": "Identity theorem in several variables",
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