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  "body": "\n## Statement\n\nLet $X$, $Y$ and $Z$ be [[topological spaces|0CNQXVBHCX]].\nLet $f \\colon X \\to Y$ and $g \\colon Y \\to Z$ be [[continuous maps|0SY9ECQ19Q]].\nThen the [[composition|07HF9YCWPP]] $g \\circ f \\colon X \\to Z$ is continuous.\n\n## Remarks\n\nProof: for each open set $W$ of $Z$, $(g \\circ f)^{-1}(W) = f^{-1}(g^{-1}(W))$. The set $g^{-1}(W)$ is open in $Y$, so its preimage under $f$ is open in $X$.\nFor example, $x \\mapsto \\sin(x^2)$ is continuous on $\\mathbb{R}$, because $x \\mapsto x^2$ and $\\sin$ are continuous.\n\n## Explanations\n\n### L1 human\n\nIf you do one continuous operation and then another one, the result is continuous.\nSmall changes in the input give small changes after the first step, and then small changes after the second step.\n",
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