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  "body": "\n## Definition\n\nLet $k$ be an algebraically closed field.\n\n1. Let $X$ be a quasi-affine variety, that is, a nonempty open subset of an [[affine variety|JERVQT4C12]] in $k^n$. A function $f \\colon X \\to k$ is *regular at* $p \\in X$ if the following holds. There are an [[open|02BH3MCBQB]] neighbourhood $U \\subseteq X$ of $p$ and polynomials $g, h \\in k[x_1, \\dots, x_n]$ such that $h$ has no zero on $U$ and $f = g/h$ on $U$.\n2. Let $X$ be a [[quasi-projective variety|EQSHJPE7H8]] in $\\mathbb{P}^n$. A function $f \\colon X \\to k$ is *regular at* $p \\in X$ if the following holds. There are an open neighbourhood $U \\subseteq X$ of $p$ and [[homogeneous polynomials|2C3PAQ7H50]] $g, h \\in k[x_0, \\dots, x_n]$ of the same degree. The polynomial $h$ has no zero on $U$, and $f = g/h$ on $U$.\n\nIn both cases, $f$ is *regular* on $X$ if it is regular at every point of $X$.\n\n## Remarks\n\nIn case 2, $g/h$ is a well-defined function on $U$, because $g$ and $h$ have the same degree $d$ and so $g(\\lambda a)/h(\\lambda a) = g(a)/h(a)$.\nA regular function is [[continuous|0VGFHTM75Y]] when $k$ is identified with $\\mathbb{A}^1$ with its Zariski topology.\n\nExample: on $X = k \\setminus \\set{0}$, the function $f(x) = 1/x$ is regular, with $g = 1$ and $h = x$ on $U = X$.\n\nNon-example: the function on $k$ given by $f(0) = 1$ and $f(x) = 0$ for $x \\ne 0$ is not regular at $0$.\nNear $0$ it would equal $g/h$ on a nonempty open set, so $g$ would vanish on a cofinite set and hence be $0$, which contradicts $f(0) = 1$.\n\nOn an affine variety $X$, the ring of regular functions on all of $X$ equals the [[coordinate ring|G9WB27VWP7]] $k[X]$; this uses Hilbert's Nullstellensatz.\nOn a [[projective variety|A4H4CZ2V35]], every regular function on the whole variety is constant.\n",
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