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  "body": "\n## Statement\n\nLet $I$ be an interval and let $f \\colon I \\to \\mathbb{R}$ be monotone.\nThen the set of points of $I$ where $f$ is not continuous is countable.\n\n## Remarks\n\nAssume $f$ increasing. By [[the existence of one-sided limits|0KJXHN0TW0]], each discontinuity $c$ inside $I$ gives an open interval $(f(c^-), f(c^+))$.\nThese intervals are disjoint. Each contains a rational number, so there are only countably many of them.\nAll these discontinuities are [[jumps|05B0FKFXQA]].\nMathlib states it as `Monotone.countable_not_continuousAt`.\n\n## Explanations\n\n### L1 human\n\nEach discontinuity of an increasing function is a jump up.\nThe jumps take disjoint pieces of the vertical axis, so there can be only countably many.\n",
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