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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, let $f \\colon U \\to \\mathbb{R}^m$, and let $a \\in U$.\nThe function $f$ is strictly differentiable at $a$ if there is a linear map $L \\colon \\mathbb{R}^n \\to \\mathbb{R}^m$ such that\n$$\\lim_{\\substack{(x, y) \\to (a, a) \\\\ x \\ne y}} \\frac{\\|f(x) - f(y) - L(x - y)\\|}{\\|x - y\\|} = 0.$$\n\n## Remarks\n\nIf $f$ is strictly differentiable at $a$, put $y = a$: then $f$ is [[differentiable|0135XZZQ8M]] at $a$ and $L = Df(a)$ is its [[Fréchet derivative|08EVNYZF42]].\nThe converse is false. The function $f(x) = x^2 \\sin(1/x)$ for $x \\ne 0$, $f(0) = 0$, is differentiable at $0$, but it is not strictly differentiable at $0$ because $f'$ is not continuous at $0$.\nA function of class $C^1$ on $U$ is [[strictly differentiable|0STPPDXGME]] at each point of $U$.\nThe same definition works for maps between Banach spaces, with norms in place of Euclidean norms.\n\n## Explanations\n\n### L1 human\n\nOrdinary differentiability compares $f(x)$ with $f(a)$ only.\nStrict differentiability compares $f(x)$ with $f(y)$ for all pairs of points near $a$.\nSo the linear approximation is uniform in a small neighbourhood, not only at the single point.\n",
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