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  "body": "\n## Definition\n\nLet $f \\colon \\mathbb{R} \\to \\mathbb{R}$ be a function and let $T$ be a real number.\nThe number $T$ is a period of $f$ if $f(x + T) = f(x)$ for all $x \\in \\mathbb{R}$.\nThe function $f$ is periodic if it has a period $T \\ne 0$.\n\n## Remarks\n\nIf $T$ is a period, then so is $k T$ for each integer $k$.\nThe [[sine function is periodic|0P4NQD746K]] with period $2 \\pi$.\nIf a continuous periodic function is not constant, it has a smallest positive period. The [[Dirichlet function|0T4EJ8ZJM9]] has every rational number as a period.\nMathlib defines periods as `Function.Periodic`.\n\n## Explanations\n\n### L1 human\n\nA periodic function repeats itself after each step of length $T$.\nIts graph is made of copies of one piece, placed side by side.\n",
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