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  "body": "\n## Statement\n\nLet $X$ be a [[paracompact|8R29G6NG9Q]] [[Hausdorff space|0S1FJ8VEX9]] and let $Y$ be a [[Banach space|06GZ956657]].\nLet $F$ be a [[multivalued function|03CPD1HGF3]] from $X$ to $Y$ such that:\n\n1. $F$ is [[lower hemicontinuous|XNMKX7H4FD]].\n2. For every $x \\in X$, the set $F(x)$ is nonempty, [[convex|02AS45T3H9]] and [[closed|0ZCK8DQYN1]] in $Y$.\n\nThen there is a [[continuous|0VGFHTM75Y]] map $f \\colon X \\to Y$ with $f(x) \\in F(x)$ for every $x \\in X$.\n\n## Remarks\n\nSuch a map $f$ is a *continuous selection* of $F$.\nThe proof first builds, for each $\\varepsilon \u003e 0$, a continuous map that lies within distance $\\varepsilon$ of $F(x)$ at every $x$.\nIt glues local choices with a partition of unity, which exists because $X$ is paracompact.\nConvexity keeps the convex combinations close to $F(x)$.\nRepeating this with $\\varepsilon = 2^{-n}$, each step close to the previous one, gives a uniformly Cauchy sequence of continuous maps.\nIts limit exists since $Y$ is complete, it is continuous, and it lies in $F(x)$ since $F(x)$ is closed.\n\nExample: let $T \\colon Y \\to Z$ be a continuous linear surjection between Banach spaces and set $F(z) = T^{-1}(z)$.\nThe values are closed affine subspaces, and the open mapping theorem makes $F$ lower hemicontinuous.\nSo $T$ has a continuous, possibly nonlinear, right inverse.\n\nLower hemicontinuity cannot be replaced by upper hemicontinuity.\nOn $X = \\R$, let $F(x) = \\set{-1}$ for $x \u003c 0$, $F(0) = [-1, 1]$ and $F(x) = \\set{1}$ for $x \u003e 0$.\nThis $F$ is upper hemicontinuous with nonempty closed convex values, but a selection must jump from $-1$ to $1$ at $0$, so no continuous selection exists.\n\n## Explanations\n\n### L1 human\n\nSuppose that at every point of a space you are given a set of allowed values, and the sets change in a reasonable way.\nThe theorem says that you can choose one allowed value at each point so that the choice varies continuously.\nThis needs the allowed sets to be convex and closed, the space of values to be a complete normed space, and the sets not to shrink suddenly.\n\n### L2 human\n\nThe theorem is the main tool to turn set-valued information into continuous functions.\nIt gives continuous right inverses of linear surjections between Banach spaces, and it is used in the theory of [[absolute neighborhood retracts|DK3MT3HYX5]] and in fixed-point and control theory.\nParacompactness enters only through partitions of unity, in the same way as in the proof of the [[Dugundji extension theorem|RBDJBK63HX]].\nCompare with the fixed-point theorems for upper hemicontinuous maps, such as Kakutani's, where the opposite kind of hemicontinuity is the right one.\n",
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