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  "body": "\n## Statement\n\nLet $J$ be an interval, let $K \\ge 0$, and let $f : J \\times \\mathbb{R}^n \\to \\mathbb{R}^n$ be continuous.\nAssume that $\\|f(t, x) - f(t, y)\\| \\le K \\|x - y\\|$ for all $t \\in J$ and $x, y \\in \\mathbb{R}^n$.\nThen for each $t_0 \\in J$ and $x_0 \\in \\mathbb{R}^n$, the [[initial value problem|075VBY4K4X]] $x' = f(t, x)$, $x(t_0) = x_0$ has exactly one solution defined on all of $J$.\n\n## Remarks\n\nThe point is that the solution does not blow up: the Lipschitz bound gives $\\|f(t, x)\\| \\le \\|f(t, 0)\\| + K \\|x\\|$, so $\\|x(t)\\|$ grows at most exponentially.\nA linear system $x' = A(t) x + b(t)$ with continuous $A$ and $b$ on $J$ satisfies the hypothesis on each compact subinterval. So its solutions exist on all of $J$.\nA field that is only locally Lipschitz can give [[blow-up in finite time|0E4GXX7SJP]].\n\n## Explanations\n\n### L1 human\n\nIf the field grows at most linearly in the state, solutions cannot run away to infinity in finite time.\nSo the solution exists as long as the field is defined.\n",
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