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  "body": "\n## Statement\n\nLet $I$ be an open interval and let $f \\colon I \\to \\mathbb{R}$ be differentiable with $f'(x) = 0$ for all $x \\in I$.\nThen $f$ is constant on $I$.\n\n## Remarks\n\nFor $x \u003c y$ in $I$, the [[mean value theorem|0BARQY52N5]] gives $f(y) - f(x) = f'(c)(y - x) = 0$.\nThe domain must be an interval. On $(0, 1) \\cup (2, 3)$, a function equal to $0$ on one part and $1$ on the other has zero derivative.\nSo two antiderivatives of a function on an interval differ by a constant. Mathlib states the result as `is_const_of_deriv_eq_zero`.\n\n## Explanations\n\n### L1 human\n\nA function whose slope is zero everywhere on an interval never rises or falls.\nSo its graph is a horizontal line.\n",
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