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  "body": "\n## Example\n\nEvery irrational $x \\in (0, 1)$ has a unique infinite [[continued fraction|0BK6JBK58W]] expansion $x = [0; a_1, a_2, a_3, \\dots]$ with positive integers $a_k$.\nThe question-mark function $? \\colon [0, 1] \\to [0, 1]$ is defined at such $x$ by\n$$?(x) = 2 \\sum_{k=1}^{\\infty} (-1)^{k+1} \\, 2^{-(a_1 + a_2 + \\cdots + a_k)}.$$\nFor a rational $x = [0; a_1, \\dots, a_n]$ the same formula with a finite sum is used, and $?(0) = 0$.\nThe value does not depend on which of the two finite expansions of a rational is chosen.\nFor example, $\\frac12 = [0; 2] = [0; 1, 1]$ gives $2 \\cdot 2^{-2} = \\frac12$ and $2(2^{-1} - 2^{-2}) = \\frac12$.\n\nSome values: $?(1) = 1$, $?(\\frac13) = 2 \\cdot 2^{-3} = \\frac14$, and for $x = [0; 1, 1, 1, \\dots] = (\\sqrt5 - 1)/2$ the geometric series gives $?(x) = 2 \\cdot \\frac{1/2}{1 + 1/2} = \\frac23$.\n\nThe function $?$ is continuous and strictly increasing, so it is a [[homeomorphism|05GZ11PWRY]] of $[0, 1]$ onto itself.\nIts derivative exists and equals $0$ at almost every point of $[0, 1]$.\nSo $?$ is a [[singular function|P5AWRD2793]].\nUnlike the [[Cantor function|NPBYYK63FE]], it is strictly increasing, so it is constant on no interval.\n\nThe function maps the rationals of $[0, 1]$ onto the dyadic rationals $m/2^n$ of $[0, 1]$.\nIt maps the quadratic irrationals of $(0, 1)$ onto the rationals that are not dyadic.\n\n## Explanations\n\n### L1 human\n\nThe question-mark function reads the continued fraction of a number and turns it into a binary expansion: each partial quotient becomes a run of equal binary digits.\nThe first run has length $a_1 - 1$, and the $k$-th run has length $a_k$ for $k \\ge 2$.\nIt is a continuous, strictly increasing function that still has slope zero almost everywhere.\nThis shows that a function can climb steadily from $0$ to $1$ while being flat in the sense of derivatives almost everywhere.\n",
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