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  "body": "\n## Definition\n\nLet $I \\subseteq \\R$ be an open interval, let $f \\colon I \\to \\R$, and let $x \\in I$.\nThe four *Dini derivatives* of $f$ at $x$ are the following elements of the [[extended real line|0GSZ7B0CGR]].\n\n1. The upper right Dini derivative $D^+ f(x) = \\limsup_{h \\to 0^+} \\frac{f(x + h) - f(x)}{h}$.\n2. The lower right Dini derivative $D_+ f(x) = \\liminf_{h \\to 0^+} \\frac{f(x + h) - f(x)}{h}$.\n3. The upper left Dini derivative $D^- f(x) = \\limsup_{h \\to 0^-} \\frac{f(x + h) - f(x)}{h}$.\n4. The lower left Dini derivative $D_- f(x) = \\liminf_{h \\to 0^-} \\frac{f(x + h) - f(x)}{h}$.\n\nHere the upper and lower limits are the [[limit superior|0P8BA7T7S5]] and [[limit inferior|0H4R3RESGV]] of the quotient.\nThey are taken over $h$ in the given one-sided range with $x + h \\in I$.\n\n## Remarks\n\nAlways $D_+ f(x) \\le D^+ f(x)$ and $D_- f(x) \\le D^- f(x)$.\nThe [[right derivative|0HXEX0DPRA]] of $f$ at $x$ exists and is finite exactly when $D_+ f(x) = D^+ f(x)$ is finite, and similarly on the left.\nSo $f$ is differentiable at $x$ exactly when all four Dini derivatives are equal and finite.\n\nExample: let $f(x) = x \\sin(1/x)$ for $x \\ne 0$ and $f(0) = 0$.\nFor $h \\ne 0$, the [[difference quotient|AZBDB5F4M4]] at $0$ is $\\frac{h \\sin(1/h)}{h} = \\sin(1/h)$.\nFor $h \u003e 0$ and for $h \u003c 0$ alike, it takes every value in $[-1, 1]$ for arbitrarily small $\\abs{h}$.\nSo $D^+ f(0) = D^- f(0) = 1$ and $D_+ f(0) = D_- f(0) = -1$, and $f$ is not differentiable at $0$.\n\nNon-example of finiteness: for $f(x) = \\sqrt{\\abs{x}}$ at $0$, the quotient for $h \u003e 0$ is $h^{-1/2}$, so $D^+ f(0) = D_+ f(0) = +\\infty$.\nThe Dini derivatives exist but are not real numbers.\n",
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