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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $X$ and $Y$ be [[normed vector spaces|03J5RAKB30]] over $\\mathbb{K}$.\nLet $U \\subseteq X$ be open, let $f \\colon U \\to Y$, let $x \\in U$ and let $h \\in X$.\nThe *Gateaux differential* of $f$ at $x$ in the direction $h$ is the limit\n$$df(x; h) = \\lim_{t \\to 0} \\frac{f(x + t h) - f(x)}{t},$$\ntaken over $t \\in \\mathbb{K} \\setminus \\set{0}$, when this limit exists in $Y$.\nThe map $f$ is *Gateaux differentiable* at $x$ if $df(x; h)$ exists for every $h \\in X$ and $h \\mapsto df(x; h)$ is a [[continuous linear map|03TM135G0N]] $X \\to Y$.\nThis map is the *Gateaux derivative* of $f$ at $x$.\n\n## Remarks\n\nFor $X = \\R^n$ and $Y = \\R$, $df(x; h)$ is the [[directional derivative|06KWGHDN2J]] of $f$ along $h$.\nSome authors do not ask $h \\mapsto df(x; h)$ to be linear or continuous.\nThe definition works without change for locally convex spaces.\nIf $f$ has a [[Fréchet derivative|06EH97V2A7]] $Df(x)$, then $f$ is Gateaux differentiable at $x$ with $df(x; h) = Df(x) h$.\n\nExample: let $H$ be a real Hilbert space and $f(x) = \\norm{x}^2$.\nThen $f(x + t h) = \\norm{x}^2 + 2t \\ip{x}{h} + t^2 \\norm{h}^2$, so $df(x; h) = 2\\ip{x}{h}$, which is continuous and linear in $h$.\n\nNon-example: on $\\R^2$, let $f(x, y) = x^3 / (x^2 + y^2)$ and $f(0, 0) = 0$.\nFor $h = (a, b) \\neq 0$, $f(t a, t b) / t = a^3 / (a^2 + b^2)$, so $df(0; h)$ exists for every $h$.\nBut $h \\mapsto a^3 / (a^2 + b^2)$ is not linear, so $f$ is not Gateaux differentiable at $0$.\n\nA Gateaux differentiable map need not be Fréchet differentiable.\nOn $\\R^2$, let $g(x, y) = 1$ when $y = x^2$ and $(x, y) \\neq 0$, and $g = 0$ elsewhere.\nEach line through $0$ meets the parabola in at most one point other than $0$, so $dg(0; h) = 0$ for all $h$.\nBut $g$ is not continuous at $0$.\n",
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