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  "body": "\n## Statement\n\nLet $U$ be an open subset of $\\mathbb{R}^n$, let $f : U \\to \\mathbb{R}^n$ be continuously differentiable, let $x_0 \\in U$ and let $t_0 \\in \\mathbb{R}$.\nThen there are real numbers $\\varepsilon \u003e 0$, $a \u003e 0$ and $L, K \\ge 0$ with these properties.\nThe closed ball $B$ of radius $a$ around $x_0$ lies in $U$.\nThe time-independent field $(t, x) \\mapsto f(x)$, extended in any way outside $B$, satisfies the [[Picard–Lindelöf hypotheses|0J7HF2PF1H]] at $(t_0, x_0)$ on $[t_0 - \\varepsilon, t_0 + \\varepsilon]$ with constants $a, L, K$.\n\n## Remarks\n\nProof: choose $a \u003e 0$ with $B \\subseteq U$. The continuous functions $\\|f\\|$ and $\\|f'\\|$ are bounded on the compact ball $B$, by $L$ and $K$. The mean value inequality makes $f$ $K$-Lipschitz on $B$. Then choose $\\varepsilon$ with $L \\varepsilon \\le a$.\nSo the initial value problem $x' = f(x)$, $x(t_0) = x_0$ has a unique local solution for each $C^1$ field $f$.\nThe same argument works for a time-dependent field $f(t, x)$ of class $C^1$.\n\n## Explanations\n\n### L1 human\n\nA smooth field has a bounded derivative near each point, so it is Lipschitz near this point.\nThis is why, in practice, one rarely checks the Lipschitz condition directly: smoothness is enough.\n",
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