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  "body": "\n## Statement\n\nLet $t_0 \\in \\R$, $x_0 \\in \\R^n$ and $a, b \u003e 0$, and let $f$ be continuous on $R = [t_0, t_0 + a] \\times \\set{x : \\norm{x - x_0} \\le b}$.\nLet $M = \\max_R \\norm{f} \u003e 0$ and $\\alpha = \\min(a, b/M)$.\nFor $c \\in [t_0, t_0 + \\alpha]$, let $S_c$ be the set of values $x(c)$ of all solutions $x : [t_0, t_0 + \\alpha] \\to \\R^n$ of $x' = f(t, x)$ with $x(t_0) = x_0$.\nThen $S_c$ is a [[continuum|HG4S5HNWXF]]: a nonempty, compact and connected subset of $\\R^n$.\n\n## Remarks\n\n$S_c$ is nonempty by the [[Peano existence theorem|098YP831QY]].\nCompactness follows from the Arzelà–Ascoli theorem, since all solutions are bounded and $M$-Lipschitz.\nConnectedness is the heart of the proof: one approximates $f$ by functions for which solutions are unique.\nFor these the value at time $c$ depends continuously on parameters, and one passes to the limit.\nExample: $x' = 2\\sqrt{\\abs{x}}$, $x(0) = 0$ has the solutions $x = 0$ and, for each $s \\ge 0$, $x(t) = 0$ for $t \\le s$ and $x(t) = (t - s)^2$ for $t \\ge s$.\nSo $S_c = [0, c^2]$, an interval, as the theorem predicts.\nIn particular, the [[initial value problem|075VBY4K4X]] has either exactly one solution on $[t_0, t_0 + \\alpha]$, or some $S_c$ has more than one point and is then uncountable.\n\n## Explanations\n\n### L1 human\n\nWhen an initial value problem has many solutions, their positions at a later time do not scatter in separate pieces.\nThey fill a connected, closed and bounded region, the funnel of solutions.\nSo non-uniqueness, when it happens, comes as a whole continuum of solutions.\n\n### L2 human\n\nThe theorem refines the [[Peano existence theorem|098YP831QY]]: it describes the whole set of solutions, not just one.\nA stronger form says that the set of solutions itself, as a subset of the space of continuous functions, is compact and connected.\nThese results are used in the theory of differential inclusions and in the study of non-unique flows.\n",
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