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  "body": "\n## Definition\n\nLet $I$ be an open [[interval|0TK8Y3J5RW]] of $\\R$, let $f \\colon I \\to \\R$ be a [[convex function|0VV0HSW1ZS]], and let $x_0 \\in I$.\nA real number $c$ is a *subderivative* of $f$ at $x_0$ if\n$$f(x) - f(x_0) \\ge c \\, (x - x_0) \\quad \\text{for all } x \\in I.$$\nThe *subdifferential* $\\partial f(x_0)$ is the set of all subderivatives of $f$ at $x_0$.\n\n## Remarks\n\nThe subdifferential is the closed interval $\\partial f(x_0) = [f'_-(x_0), f'_+(x_0)]$, where $f'_-$ and $f'_+$ are the one-sided derivatives, which exist for a convex function on an open interval.\nIn particular it is never empty.\nIt is the single point $\\set{f'(x_0)}$ exactly when $f$ has a [[derivative|0F1WR9C83K]] at $x_0$.\nFor a convex function on $\\R^n$, a vector $g$ with $f(x) - f(x_0) \\ge \\ip{g}{x - x_0}$ for all $x$ is called a *subgradient*.\n\nExample: $f(x) = \\abs{x}$ on $\\R$ and $x_0 = 0$.\nFor $c \\in [-1, 1]$ one has $\\abs{x} \\ge c x$ for all $x$, so $\\partial f(0) = [-1, 1]$.\n\nNon-example: $c = 2$ for the same $f$ and $x_0$.\nAt $x = 1$ the condition reads $1 \\ge 2$, which fails.\n",
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