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  "body": "\n## Summary\n\nThe Dini derivatives of a real function at a point are the four one-sided upper and lower limits of its [[difference quotients|AZBDB5F4M4]].\nUnlike the [[derivative|0F1WR9C83K]], they always exist, possibly as $+\\infty$ or $-\\infty$.\nA function is differentiable at a point exactly when its four Dini derivatives there are equal and finite.\nThey are the main tool for proving differentiability almost everywhere, as in [[Lebesgue's theorem on monotone functions|1Y4N1DDQNR]].\n\n## Explanations\n\n### L1 human\n\nAt a corner or a wiggle, the slope of a function may fail to have a limit.\nDini derivatives still record what happens: the largest and smallest slopes seen from the right, and the largest and smallest seen from the left.\nFor $\\abs{x}$ at $0$, the slopes from the right are all $1$ and the slopes from the left are all $-1$.\nSo the right Dini derivatives are $1$ and the left ones are $-1$.\n",
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