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  "body": "\n## Summary\n\nThe limiting subdifferential, also called the Mordukhovich or basic subdifferential, is a set of generalized gradients of a function at a point.\nIt is built in two steps.\nOne first takes the regular (Fréchet) subgradients at nearby points, then all limits of these as the points approach the given point.\nFor a convex function it equals the [[subdifferential|9C5WXYYQEK]] of convex analysis, and for a locally Lipschitz function its convex hull is the [[Clarke generalized gradient|20967BXZKA]].\nIt is usually smaller than the Clarke generalized gradient and has a good calculus, which makes it a main tool of variational analysis.\nThe [[definition|YXJDQM9VVD]] gives the precise construction.\n\n## Explanations\n\n### L1 human\n\nAt a smooth point, the gradient is the only reasonable slope.\nAt a kink, one looks at the slopes that work just beside the point and keeps all their limits.\nFor $-\\abs{x}$ at $0$, no slope works at the peak itself, but the nearby slopes are $1$ and $-1$.\nSo the limiting subdifferential is the two-point set $\\set{-1, 1}$, while the Clarke generalized gradient fills in the whole interval $[-1, 1]$.\n",
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