Timeline for If $f$ is infinitely differentiable then $f$ coincides with a polynomial
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| May 2, 2023 at 19:54 | comment | added | Aaron Hill | Should $x: \forall (a,b)\ni x$ instead be $x: \exists(a,b)\ni x$ ? If $f\restriction_{(a, b)}$ is required to hold for every interval $(a, b)$, then $X$ would need to be either empty or the entire interval $[0, 1]$. | |
| May 2, 2020 at 16:44 | comment | added | MathematicsStudent1122 | @AlexW You're right, thank you! | |
| May 2, 2020 at 12:04 | comment | added | Alex W | @MathematicsStudent1122 Let $f(x_1,x_2)=e^{x_2}$, $\alpha=(1,0)$. Then $f^{(\alpha)}(x)=0$ for all $x\in\mathbb{R}^2$ but $f$ is not a polynomial. | |
| May 1, 2020 at 16:02 | comment | added | MathematicsStudent1122 | @AlexW In that case, the natural hypothesis would be that for each $x_0 \in \mathbb{R}^n$ there exists a multi-index $\alpha=\alpha(x_0)$ such that $\left. \frac{\partial^{|\alpha|} f}{\partial x_1^{\alpha_1} \partial x_2^{\alpha_2} \cdots \partial x_n^{\alpha_n}}\right\vert_{x_0}=0$. Essentially the same argument works, by appropriately replacing intervals $(a,b)$ as in this answer with open balls. In particular there are only countably many multi-indices so Baire's theorem applies. | |
| Jan 14, 2019 at 17:40 | comment | added | Alex W | Is a similar statement true for functions of n>1 variables? | |
| Nov 1, 2011 at 11:19 | history | edited | Emil Jeřábek | CC BY-SA 3.0 |
fix TeX
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| May 8, 2011 at 22:48 | comment | added | Joshua P. Swanson | Thank you! Filling in all the details to this outline is a fantastic exercise in basic real analysis and topology. It strikes me as a great "capstone" to a relevant course. It went through at least 20 relevant topics/ideas: (in roughly decreasing order of complexity) Baire Category Theorem, Heine-Borel, infs/sups (so LUB property of R), compactness, Cauchy/convergent sequences/completeness, (infinite) differentiability, continuity, connectedness, perfect sets, limit points (from the sides), induction, isolated points, open/closed sets, interiors, derivatives of polynomials, and boundedness. | |
| Sep 6, 2010 at 14:51 | vote | accept | C.S. | ||
| Jul 31, 2010 at 23:57 | history | edited | Andrey Gogolev | CC BY-SA 2.5 |
added 101 characters in body
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| Jul 31, 2010 at 23:20 | history | answered | Andrey Gogolev | CC BY-SA 2.5 |