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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{C}$ be an open set and let $f : U \\to \\mathbb{C}$ be continuous.\nAssume that $\\oint_{\\partial T} f(z) \\, dz = 0$ for each closed triangle $T \\subseteq U$, where $\\partial T$ is the boundary of $T$ traveled once and the integral is a [[contour integral|00FWNP9TXY]].\nThen $f$ is [[holomorphic|0DG0Z72CQD]] on $U$.\n\n## Remarks\n\nIt is enough to know that $\\oint_\\gamma f(z) \\, dz = 0$ for each closed piecewise continuously differentiable path $\\gamma$ in $U$, because the boundary of a triangle is such a path.\nProof idea: on each disc in $U$, $F(z) = \\int_{[c, z]} f(w) \\, dw$ is a primitive of $f$. So $F$ is holomorphic, and $f = F'$ is holomorphic because [[derivatives of holomorphic functions are holomorphic|0QEBN8DGVE]].\nTogether with [[Goursat's theorem|0BRG85WXYC]], this shows: a continuous $f$ is holomorphic on $U$ if and only if its integrals over all triangles in $U$ are zero.\nA main use is the theorem on [[uniform limits of holomorphic functions|0431RN0R1N]].\n\n## Explanations\n\n### L1 human\n\nMorera's theorem is a converse of Cauchy's theorem.\nIf the integral of a continuous function around each small triangle is zero, then the function is holomorphic.\n",
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