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  "body": "\n## Statement\n\nLet $D \\subseteq \\C$ be an open set, let $a, b \\in D$, and let $f$ be a [[holomorphic function|0DG0Z72CQD]] on a disc around $a$.\nLet $\\gamma_0$ and $\\gamma_1$ be [[paths|0JDE56HWQ8]] in $D$ from $a$ to $b$.\nLet $H \\colon [0, 1] \\times [0, 1] \\to D$ be a homotopy from $\\gamma_0$ to $\\gamma_1$ with fixed end points, and write $\\gamma_s = H(s, \\cdot)$.\nAssume that $f$ has an [[analytic continuation|0M3YQJ81NB]] along every path $\\gamma_s$, $0 \\le s \\le 1$.\nThen the continuations of $f$ along $\\gamma_0$ and along $\\gamma_1$ agree near $b$.\n\n## Remarks\n\nThe idea of the proof is that the continuation along $\\gamma_s$ depends continuously on $s$.\nFor $s'$ close to $s$, the two paths stay close, and the continuations along them use overlapping discs, so they give the same germ at $b$.\nCompactness of $[0, 1]$ then passes from $s = 0$ to $s = 1$.\nAn important consequence concerns a [[simply connected domain|075C629XTH]] $D$.\nIf $f$ can be continued along every path in $D$ that starts at $a$, then there is a holomorphic function $F$ on $D$ that equals $f$ near $a$.\nThe homotopy hypothesis is needed.\nOn $D = \\C \\setminus \\set{0}$, a branch of the [[complex logarithm|0X2FY491PF]] near $a = 1$ continues along every path, but continuation along the unit circle returns to $1$ with the value increased by $2 \\pi i$.\nThe unit circle is not homotopic to a constant path in $D$, so there is no contradiction.\n\n## Explanations\n\n### L1 human\n\nAnalytic continuation extends a function step by step along a path.\nIn general, two paths to the same point can give two different values, as for the logarithm going around $0$.\nThe monodromy theorem says this cannot happen when one path can be deformed into the other without leaving the region where continuation works.\nSo on a region without holes, continuation produces a single-valued function.\n\n### L2 human\n\nThe theorem is the standard way to construct functions on simply connected domains, such as logarithms and square roots of zero-free holomorphic functions.\nIt is the analytic form of a topological fact: a covering space over a simply connected space is trivial.\nSeen in this way, the continuations of a germ form a Riemann surface spread over the plane.\nThe theorem says that this surface has only one sheet over a simply connected region.\nThe monodromy group measures the failure over regions that are not simply connected.\n",
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