{
  "aliases": [
    "principle-of-isolated-zeros",
    "zeros-are-isolated"
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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{C}$ be an open set, let $f : U \\to \\mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$, and let $z_0 \\in U$.\nThen exactly one of these two cases holds.\n\n1. There is $r \u003e 0$ such that $f(z) = 0$ for each $z$ with $|z - z_0| \u003c r$.\n2. There is $r \u003e 0$ such that $f(z) \\neq 0$ for each $z$ with $0 \u003c |z - z_0| \u003c r$.\n\n## Remarks\n\nProof: if case 1 does not hold, then $f(z) = (z - z_0)^m g(z)$ near $z_0$ with $g(z_0) \\neq 0$, where $m$ is the [[order|064N002TX4]] of $f$ at $z_0$. By continuity, $g \\neq 0$ near $z_0$.\nOn a [[domain|097KNJFJ7V]], it follows that the zeros of a holomorphic function that is not identically zero have no accumulation point in the domain.\nThe real function $x \\mapsto e^{-1/x^2}$, with value $0$ at $0$, is infinitely differentiable and has all derivatives $0$ at $0$. The theorem has no analogue for such real functions.\n\n## Explanations\n\n### L1 human\n\nNear a point, a holomorphic function is either zero everywhere or zero only at that point.\nSo the zeros of a nonzero holomorphic function cannot pile up inside its domain, unlike the zeros of $\\sin(1/x)$ near $0$.\n",
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    "aliases": [
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    "name": "Isolated zeros theorem",
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        "url": "https://en.wikipedia.org/wiki/Identity_theorem"
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  "markdown": "https://math2.tamnd.com/node/0K83KEJ43P.md",
  "msc": [
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  "name": "Isolated zeros theorem",
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  "role": "theorem",
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