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  "body": "\n## Definition\n\nLet $X$ be a real [[Banach space|06GZ956657]] with [[continuous dual space|4A82S5FVRT]] $X^*$, and write $\\ip{\\zeta}{v} = \\zeta(v)$.\nLet $f \\colon X \\to \\R$ be [[Lipschitz|0AW1E6YTJY]] on a neighbourhood of a point $x \\in X$.\nThe *Clarke generalized directional derivative* of $f$ at $x$ in the direction $v \\in X$ is\n$$f^\\circ(x; v) = \\limsup_{y \\to x, \\ t \\downarrow 0} \\frac{f(y + t v) - f(y)}{t},$$\na [[limit superior|0T3RZQXDYD]] over $y \\in X$ and $t \u003e 0$.\nThe *Clarke generalized gradient* of $f$ at $x$ is the set\n$$\\partial_C f(x) = \\set{\\zeta \\in X^* : \\ip{\\zeta}{v} \\le f^\\circ(x; v) \\text{ for all } v \\in X}.$$\n\n## Remarks\n\nIf $K$ is a Lipschitz constant of $f$ near $x$, then $\\abs{f^\\circ(x; v)} \\le K \\norm{v}$, and $v \\mapsto f^\\circ(x; v)$ is sublinear.\nSo $\\partial_C f(x)$ is convex, nonempty by the Hahn–Banach theorem, and weak-* compact by the Banach–Alaoglu theorem, since it is weak-* closed and bounded by $K$.\nIf $f$ is continuously differentiable near $x$, then $\\partial_C f(x) = \\set{f'(x)}$.\nIf $f$ is convex, then $\\partial_C f(x)$ is the subdifferential of convex analysis.\nFor $X = \\R^n$, Clarke's formula describes it through gradients, which exist almost everywhere by Rademacher's theorem.\nIt is the convex hull of the limits $\\lim_k \\nabla f(x_k)$ over sequences $x_k \\to x$ at which $f$ is differentiable.\nThis formula still holds when the $x_k$ must avoid a given set of measure zero.\n\nExample: $f(x) = -\\abs{x}$ on $\\R$ at $x = 0$.\nFor $y \u003e 0$ small the difference quotient in direction $v$ is $-v$, and for $y \u003c 0$ small it is $v$, so $f^\\circ(0; v) = \\abs{v}$.\nHence $\\partial_C f(0) = \\set{\\zeta : \\zeta v \\le \\abs{v} \\text{ for all } v} = [-1, 1]$.\n\nNon-example: $f(x) = \\sqrt{\\abs{x}}$ on $\\R$ is not Lipschitz near $0$.\nThe quotient $(f(t) - f(0))/t = t^{-1/2}$ is unbounded as $t \\downarrow 0$, so $f^\\circ(0; 1) = +\\infty$, and the construction gives no useful information.\n",
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