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  "body": "\n## Definition\n\nLet $(G, +)$ be an [[abelian group|082GHNPJ53]], let $(E, \\norm{\\cdot})$ be a [[normed vector space|03J5RAKB30]], and let $\\delta \\ge 0$.\nA map $f \\colon G \\to E$ is *$\\delta$-additive* if\n$$\\norm{f(x + y) - f(x) - f(y)} \\le \\delta \\quad \\text{for all } x, y \\in G.$$\nA map is *approximately additive* if it is $\\delta$-additive for some $\\delta \\ge 0$.\n\n## Remarks\n\nThe $0$-additive maps are exactly the additive maps.\nExample: $f(x) = \\floor{x}$ on $\\R$ is $1$-additive.\nWrite $x = \\floor{x} + s$ and $y = \\floor{y} + t$ with $s, t \\in [0, 1)$; then $\\floor{x + y} - \\floor{x} - \\floor{y} = \\floor{s + t} \\in \\set{0, 1}$.\nNon-example: $f(x) = x^2$ on $\\R$ is not approximately additive, since $f(x + y) - f(x) - f(y) = 2xy$ is unbounded.\nEvery map $f = A + b$ with $A$ additive and $\\norm{b(x)} \\le \\beta$ for all $x$ is $3\\beta$-additive.\n",
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