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  "body": "\n## Statement\n\nLet $U \\subseteq \\R^n$ be open and let $f \\colon U \\to \\R$ be a real [[analytic function|081WKEDK6C]].\n\n1. Let $Z = f^{-1}(0)$, assume $Z \\neq \\emptyset$, and let $K \\subseteq U$ be [[compact|0R6EWFKRCF]].\nThen there are constants $C \u003e 0$ and $\\alpha \u003e 0$ such that $\\abs{f(x)} \\ge C \\operatorname{dist}(x, Z)^{\\alpha}$ for all $x \\in K$.\n2. Let $a \\in U$.\nThen there are a neighbourhood $W \\subseteq U$ of $a$ and constants $C \u003e 0$ and $\\theta \\in (0, 1)$ such that $\\abs{f(x) - f(a)}^{\\theta} \\le C \\norm{\\nabla f(x)}$ for all $x \\in W$.\n\n## Remarks\n\nPart 1 is the distance inequality and part 2 is the gradient inequality.\nŁojasiewicz proved them with his theory of semianalytic sets.\nA later route uses resolution of singularities to make $f$ locally a monomial times a unit, where both inequalities are direct.\n\nAnalyticity is necessary.\nThe smooth function with $f(x) = e^{-1/x^2}$ for $x \\neq 0$ and $f(0) = 0$ has $Z = \\set{0}$.\nBut $\\abs{f(x)} / \\abs{x}^{\\alpha} \\to 0$ as $x \\to 0$ for every $\\alpha \u003e 0$, so part 1 fails on $K = [-1, 1]$.\n\nFor $f(x) = x^2$ on $\\R$ and $a = 0$, part 2 holds with $\\theta = 1/2$ and $C = 1/2$, since $\\abs{x^2}^{1/2} = \\abs{x} = \\tfrac12 \\abs{f'(x)}$.\n\n## Explanations\n\n### L1 human\n\nAn analytic function cannot become small without being close to its zero set: it is at least a power of the distance to its zeros.\nIn the same spirit, near a critical point the gradient cannot be much smaller than the change in the value of the function.\nBoth say that analytic functions are never infinitely flat.\n\n### L2 human\n\nThe gradient inequality is the key to the convergence of gradient flows.\nIf $f$ is real analytic and $x(t)$ is a bounded trajectory of $x' = -\\nabla f(x)$, the inequality bounds the length of the trajectory, so $x(t)$ converges to a single critical point.\nFor merely smooth functions a bounded gradient trajectory can accumulate on a whole circle of critical points.\nThe distance inequality is used to divide distributions by real analytic functions.\n",
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