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  "body": "\n## Statement\n\nLet $D \\subseteq \\mathbb{C}$ be a [[domain|097KNJFJ7V]] and let $f, g : D \\to \\mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $D$.\nLet $S = \\{z \\in D : f(z) = g(z)\\}$.\nAssume that $S$ has an accumulation point in $D$. This means that there is $z_0 \\in D$ such that each disc with center $z_0$ contains a point of $S$ other than $z_0$.\nThen $f(z) = g(z)$ for each $z \\in D$.\n\n## Remarks\n\nProof idea: apply the [[isolated zeros theorem|0K83KEJ43P]] to $f - g$. The set of points near which $f - g = 0$ is open, closed in $D$ and not empty, so it is $D$ because $D$ is connected.\nFor example, if $f$ is entire and $f(1/n) = 0$ for each $n \\ge 1$, then $f = 0$.\nThe connectedness of $D$ is necessary: on the union of two disjoint discs, a function can be $0$ on one disc and $1$ on the other.\nThe theorem makes [[analytic continuation|0M3YQJ81NB]] unique.\n\n## Explanations\n\n### L1 human\n\nTwo holomorphic functions on a domain that agree on a small piece, even on a convergent sequence of points with its limit, agree everywhere.\nSo a holomorphic function is fixed everywhere by its values on a small disc.\n",
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