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  "body": "\n## Proof\n\nWe use the slope inequalities to bound chord slopes from a fixed point, which gives a local Lipschitz bound.\n\n1. Fix $y \\in I$. Since $I$ is open, there are $x, z \\in I$ with $x \u003c y \u003c z$.\n2. For $w \\in I$ with $w \\ne y$, let $\\sigma(w) = \\dfrac{f(w) - f(y)}{w - y}$. Note that $\\sigma(w)$ is also $\\dfrac{f(y) - f(w)}{y - w}$.\n3. Let $s_- = \\dfrac{f(y) - f(x)}{y - x}$ and $s_+ = \\dfrac{f(z) - f(y)}{z - y}$.\n4. Let $y \u003c w \u003c z$. The [[slope inequalities|0BY5FTHEEE]] for $x \u003c y \u003c w$, first against last quotient, give $s_- \\le \\sigma(w)$.\n5. The slope inequalities for $y \u003c w \u003c z$, first against middle quotient, give $\\sigma(w) \\le s_+$.\n6. Let $x \u003c w \u003c y$. The slope inequalities for $x \u003c w \u003c y$, middle against last quotient, give $s_- \\le \\sigma(w)$.\n7. The slope inequalities for $w \u003c y \u003c z$, first against last quotient, give $\\sigma(w) \\le s_+$.\n8. So $s_- \\le \\sigma(w) \\le s_+$ for all $w \\in (x, z)$ with $w \\ne y$. Let $C = \\abs{s_-} + \\abs{s_+}$. Then $\\abs{\\sigma(w)} \\le C$.\n9. Hence $\\abs{f(w) - f(y)} \\le C \\abs{w - y}$ for all $w \\in (x, z)$. For $w = y$ this is trivial.\n10. Let $V$ be an open interval containing $f(y)$. Choose $\\varepsilon \u003e 0$ with $(f(y) - \\varepsilon, f(y) + \\varepsilon) \\subseteq V$.\n11. Let $\\delta = \\min\\set{y - x, \\, z - y, \\, \\varepsilon/(C + 1)}$ and $U = (y - \\delta, y + \\delta)$. Then $U \\subseteq (x, z)$.\n12. For $w \\in U$, step 9 gives $\\abs{f(w) - f(y)} \\le C \\delta \\le C \\varepsilon/(C + 1) \u003c \\varepsilon$. So $f(w) \\in V$.\n13. Thus $f$ is [[continuous at|0EEE2CZB2A]] $y$. Since $y \\in I$ was arbitrary, $f$ is [[continuous|0P27W9RW0G]] on $I$.\n\n## Idea\n\nNear an interior point $y$, every chord slope from $y$ is squeezed between two fixed slopes. They are the slopes from a point on the left and to a point on the right. Bounded slopes mean $f(w)$ is within a constant times $\\abs{w - y}$ of $f(y)$. Openness of $I$ is what gives room on both sides of $y$.\n",
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