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  "body": "\n## Definition\n\nA *Stieltjes function* is a function $F \\colon \\R \\to \\R$ such that:\n\n1. $F$ is non-decreasing, that is, $F(x) \\le F(y)$ whenever $x \\le y$; see [[monotone function|0A7KV2WB7A]].\n2. $F$ is right-continuous, that is, $F(x) = \\lim_{y \\to x^+} F(y)$ for every $x \\in \\R$.\n\n## Remarks\n\nExample: the floor function $F(x) = \\floor{x}$ is a Stieltjes function.\nIt is non-decreasing, and on each interval $[n, n + 1)$ with $n \\in \\Z$ it is constant, so it is right-continuous at every point.\nIts [[Lebesgue–Stieltjes measure|QV75XJBFCS]] puts mass $1$ at each integer.\n\nNon-example: the ceiling function $F(x) = \\ceil{x}$ is non-decreasing but is not right-continuous at the integers.\nAt $x = 0$ one has $F(0) = 0$, while $F(y) = 1$ for all $y \\in (0, 1)$, so the limit from the right is $1 \\ne F(0)$.\n\nEvery non-decreasing function has one-sided limits at every point, so right-continuity only fixes the value at the countably many jumps.\nTwo Stieltjes functions give the same measure exactly when they differ by a constant.\nSome authors write \"distribution function\" for the same notion.\n",
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