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  "body": "\n## Statement\n\nLet $J$ be an interval, let $U$ be an open subset of $\\mathbb{R}^n$, let $K \\ge 0$, and let $f : J \\times U \\to \\mathbb{R}^n$ be continuous.\nAssume that $\\|f(t, x) - f(t, y)\\| \\le K \\|x - y\\|$ for all $t \\in J$ and $x, y \\in U$.\nLet $a \u003c b$ with $[a, b] \\subseteq J$, and let $\\gamma_1, \\gamma_2 : [a, b] \\to U$ be [[solutions|04SW7WECS8]] of $x' = f(t, x)$.\nThen\n$$\\|\\gamma_1(t) - \\gamma_2(t)\\| \\le \\|\\gamma_1(a) - \\gamma_2(a)\\|\\, e^{K (t - a)} \\quad \\text{for each } t \\in [a, b].$$\n\n## Remarks\n\nSo the solution at time $t$ is a Lipschitz function of the initial value, with constant $e^{K(t - a)}$.\nThe proof applies [[Grönwall's inequality|07EJHQYR9A]] to $u(t) = \\|\\gamma_1(t) - \\gamma_2(t)\\|$.\nExistence, uniqueness and continuous dependence together make the initial value problem well posed, in the sense of Jacques Hadamard.\nFor equal initial values, the bound gives [[uniqueness|025PKW8AMR]] again.\n\n## Explanations\n\n### L1 human\n\nIf two solutions start close together, they stay close for a while.\nTheir distance can grow, but at most exponentially, with rate given by the Lipschitz constant.\nSo a small error in the initial value gives a small error in the solution over a bounded time.\n",
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