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  "body": "\n## Definition\n\nLet $I \\subseteq \\R$ be an [[interval|0TK8Y3J5RW]] and let $f \\colon I \\to \\R$ be a [[function|0XHDTE5YMZ]].\nThe function $f$ is a *Darboux function*, or has the *intermediate value property*, if the following holds.\nFor all $a \u003c b$ in $I$ and each real $y$ between $f(a)$ and $f(b)$, there is $c \\in [a, b]$ with $f(c) = y$.\n\n## Remarks\n\nEquivalently, $f$ maps every subinterval of $I$ onto an interval.\nEach [[continuous function|0P27W9RW0G]] on $I$ is a Darboux function, by the [[intermediate value theorem|00KAFBR32E]].\nEach derivative on an open interval is a Darboux function, by [[Darboux's theorem|0Q5A47J48D]].\n\nExample: let $f(x) = \\sin(1/x)$ for $x \\ne 0$ and $f(0) = 0$, on $\\R$.\nOn a subinterval not containing $0$, $f$ is continuous, so its image is an interval.\nA subinterval with more than one point that contains $0$ contains a full period of $\\sin(1/x)$ on one side of $0$, so its image is $[-1, 1]$.\nSo $f$ is a Darboux function, although it is not continuous at $0$.\n\nNon-example: the floor function $\\floor{x}$ on $\\R$ has $\\floor{0} = 0$ and $\\floor{1} = 1$, but it never takes the value $1/2$.\n\nThe [[Conway base 13 function|91YF82M8AA]] is a Darboux function that is continuous at no point.\n",
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