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  "aliases": [
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  "body": "\n## Statement\n\nLet $D$ be an open subset of $\\mathbb{R} \\times \\mathbb{R}^n$, let $f : D \\to \\mathbb{R}^n$ be continuous, and let $(t_0, x_0) \\in D$.\nThen there are an open interval $I$ with $t_0 \\in I$ and a differentiable function $z : I \\to \\mathbb{R}^n$ with these properties.\nFor each $t \\in I$, the point $(t, z(t))$ is in $D$ and $z'(t) = f(t, z(t))$.\nAlso, $z(t_0) = x_0$.\n\n## Remarks\n\nThe theorem needs only continuity of $f$. It gives existence, but not uniqueness. See the [[non-uniqueness example|00K7ASZ1PR]].\nWith a Lipschitz condition in $x$, the [[Picard–Lindelöf theorem|0P6KRD5B7A]] gives existence and uniqueness.\nA common proof builds Euler polygons with smaller and smaller steps and takes a uniformly convergent subsequence by the Arzelà–Ascoli theorem.\n\n## History\n\nGiuseppe Peano published the theorem in 1886, with a proof that was not correct.\nIn 1890 he published a correct proof that uses successive approximations.\nThe name Cauchy–Peano theorem also honours Augustin-Louis Cauchy.\n\n## Explanations\n\n### L1 human\n\nIf the vector field is continuous, at least one solution starts at each point.\nThe theorem does not say that the solution is unique, and it gives the solution only for a short time.\n",
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