Timeline for Blackbox Theorems
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Jun 15, 2012 at 16:24 | history | edited | Zsbán Ambrus | CC BY-SA 3.0 |
add Rademacher's theorem.
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| Jun 14, 2012 at 23:39 | comment | added | Paul Siegel | Perhaps you could replace your fifth example with Rademacher's theorem: every Lipschitz function from an open subset of $\mathbb{R}^n$ to $\mathbb{R}^m$ is differentiable almost everywhere. This is a more serious result which people use all the time, and I'm not sure everyone really knows the proof (though maybe I should speak for myself). | |
| Jun 14, 2012 at 21:55 | history | edited | Zsbán Ambrus | CC BY-SA 3.0 |
added 219 characters in body
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| Jun 14, 2012 at 21:44 | history | edited | Zsbán Ambrus | CC BY-SA 3.0 |
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| Jun 14, 2012 at 13:12 | comment | added | Ian Morris | Most of those are genuinely laborious proofs, but the one about convex functions can be done in a few lines. A convex function clearly has at most two intervals of monotonicity, which implies that the left and right limits at each point exist. If they aren't the same for some point then we can find a chord between two points of the graph close to the discontinuity which passes below the graph (on the left if the jump is downwards, or to the right if it is upwards) contradicting convexity. Differentiability is obtained by showing that (f(x+r)−f(x))/r is monotone in r. | |
| Jun 14, 2012 at 10:06 | history | edited | Zsbán Ambrus | CC BY-SA 3.0 |
expand other examples to a list; added 4 characters in body
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| Jun 14, 2012 at 9:30 | comment | added | Zsbán Ambrus | Uh… just because books have the proof doesn't mean we take the time to read and understand it. | |
| Jun 13, 2012 at 22:59 | comment | added | Benjamin Steinberg | This example is not really what I want because all basic algebraic topology books do it. | |
| Jun 13, 2012 at 22:26 | history | edited | Zsbán Ambrus | CC BY-SA 3.0 |
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| Jun 13, 2012 at 22:16 | history | edited | Zsbán Ambrus | CC BY-SA 3.0 |
added 188 characters in body; added 128 characters in body
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| Jun 13, 2012 at 22:08 | history | answered | Zsbán Ambrus | CC BY-SA 3.0 |