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  "body": "\n## Statement\n\nLet $D \\subseteq \\C^n$ be an open set and let $f \\colon D \\to \\C$ be a function.\nSuppose that $f$ is holomorphic in each variable separately.\nThat is, for each $j$ and each choice of the other coordinates, the function $z_j \\mapsto f(z_1, \\dots, z_n)$ is [[holomorphic|0NAHXR1FRH]] where it is defined.\nThen $f$ is a [[holomorphic function|507C34W0ZT]] on $D$; in particular, $f$ is continuous.\n\n## Remarks\n\nIf $f$ is also assumed continuous, the result is easier: it is [[Osgood's lemma|NBFNXA2WFM]].\nThe same easy argument works if $f$ is only assumed locally bounded.\nIn that case the iterated Cauchy formula on small polydiscs gives the power series of $f$.\nHartogs's proof first finds, by a Baire category argument, small polydiscs on which $f$ is bounded.\nIt then uses [[Hartogs's lemma for subharmonic functions|ZENZT8596D]] on the power series in one variable to get holomorphy at every point.\nNo such result holds for real variables.\nThe function $g(x, y) = xy/(x^2 + y^2)$ with $g(0, 0) = 0$ is real analytic in each variable separately, yet it is not continuous at the origin.\nExample: $f(z_1, z_2) = e^{z_1 z_2}$ is entire in $z_1$ for each fixed $z_2$ and in $z_2$ for each fixed $z_1$, so it is holomorphic on $\\C^2$.\n\n## Explanations\n\n### L1 human\n\nTo check that a function of several complex variables is holomorphic, it is enough to check it one variable at a time.\nNo continuity needs to be assumed; it comes for free.\nThis is special to complex variables: the same statement fails for real smooth or analytic functions.\n\n### L2 human\n\nThe theorem shows that the definition of holomorphy in several variables is very robust.\nComplex differentiability, holomorphy in each variable, local power series expansion, and the Cauchy–Riemann equations for $C^1$ functions all give the same class of functions.\nIn practice one often uses the easier Osgood version, since continuity is usually known.\nThe full theorem is a striking early result of the theory, and its proof introduced methods that reappear in the study of plurisubharmonic functions.\n",
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