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  "body": "\n## Definition\n\nLet $X$ be a set and let $f \\colon X \\to X$ be a [[map|0EPAT51DXQ]].\nA point $x \\in X$ is a *fixed point* of $f$ if $f(x) = x$.\nThe set of fixed points of $f$ is $\\operatorname{Fix}(f) = \\set{x \\in X : f(x) = x}$.\n\n## Remarks\n\nIf $x$ is a fixed point of $f$, then $f^n(x) = x$ for every $n \\ge 0$, so the orbit of $x$ is the single point $\\set{x}$.\n\nExample: $f(x) = \\cos x$ on $\\R$ has a fixed point $x_0 \\in (0, 1)$.\nIndeed $g(x) = \\cos x - x$ is continuous with $g(0) = 1 \u003e 0$ and $g(1) = \\cos 1 - 1 \u003c 0$, so $g$ vanishes somewhere in $(0, 1)$.\n\nNon-example: the translation $f(x) = x + 1$ on $\\R$ has no fixed point, since $x + 1 \\ne x$ for every $x$.\n\nFor a function of one real variable, $x$ is a fixed point exactly when the graph of $f$ meets the line $y = x$ at $(x, x)$.\nIn Lean's Mathlib the predicate is `Function.IsFixedPt f x`.\n",
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