What is a smooth function on the closure of an open set?












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Assume $U$ is an open subset of $R^{n}$. What does the notation $C^{infty}(bar{U})$ mean? For people who are familiar with sheaf, it should mean the functions on $bar{U}$ which are the restrictions of smooth functions on a neighborhood of $bar{U}$. For people familiar with PDE, it always means smooth functions on $U$ of which any derivatives(including the original function) can continuously extend to $bar{U}$. Are these two views the same? For $U$ with smooth boundary, they refer to the same class of functions. But what about general $U$?










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  • $begingroup$
    That $fin mathcal C^infty (U)$ and for all $uin partial U$ (the boundary of $U$) and all $alpha $ (multi indice), $lim_{xto u}partial _alpha f(x)$ exist.
    $endgroup$
    – Surb
    Dec 20 '18 at 10:06


















2












$begingroup$


Assume $U$ is an open subset of $R^{n}$. What does the notation $C^{infty}(bar{U})$ mean? For people who are familiar with sheaf, it should mean the functions on $bar{U}$ which are the restrictions of smooth functions on a neighborhood of $bar{U}$. For people familiar with PDE, it always means smooth functions on $U$ of which any derivatives(including the original function) can continuously extend to $bar{U}$. Are these two views the same? For $U$ with smooth boundary, they refer to the same class of functions. But what about general $U$?










share|cite|improve this question









$endgroup$












  • $begingroup$
    That $fin mathcal C^infty (U)$ and for all $uin partial U$ (the boundary of $U$) and all $alpha $ (multi indice), $lim_{xto u}partial _alpha f(x)$ exist.
    $endgroup$
    – Surb
    Dec 20 '18 at 10:06
















2












2








2





$begingroup$


Assume $U$ is an open subset of $R^{n}$. What does the notation $C^{infty}(bar{U})$ mean? For people who are familiar with sheaf, it should mean the functions on $bar{U}$ which are the restrictions of smooth functions on a neighborhood of $bar{U}$. For people familiar with PDE, it always means smooth functions on $U$ of which any derivatives(including the original function) can continuously extend to $bar{U}$. Are these two views the same? For $U$ with smooth boundary, they refer to the same class of functions. But what about general $U$?










share|cite|improve this question









$endgroup$




Assume $U$ is an open subset of $R^{n}$. What does the notation $C^{infty}(bar{U})$ mean? For people who are familiar with sheaf, it should mean the functions on $bar{U}$ which are the restrictions of smooth functions on a neighborhood of $bar{U}$. For people familiar with PDE, it always means smooth functions on $U$ of which any derivatives(including the original function) can continuously extend to $bar{U}$. Are these two views the same? For $U$ with smooth boundary, they refer to the same class of functions. But what about general $U$?







calculus derivatives pde sheaf-theory






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asked Dec 20 '18 at 10:03









殷晨曦殷晨曦

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212












  • $begingroup$
    That $fin mathcal C^infty (U)$ and for all $uin partial U$ (the boundary of $U$) and all $alpha $ (multi indice), $lim_{xto u}partial _alpha f(x)$ exist.
    $endgroup$
    – Surb
    Dec 20 '18 at 10:06




















  • $begingroup$
    That $fin mathcal C^infty (U)$ and for all $uin partial U$ (the boundary of $U$) and all $alpha $ (multi indice), $lim_{xto u}partial _alpha f(x)$ exist.
    $endgroup$
    – Surb
    Dec 20 '18 at 10:06


















$begingroup$
That $fin mathcal C^infty (U)$ and for all $uin partial U$ (the boundary of $U$) and all $alpha $ (multi indice), $lim_{xto u}partial _alpha f(x)$ exist.
$endgroup$
– Surb
Dec 20 '18 at 10:06






$begingroup$
That $fin mathcal C^infty (U)$ and for all $uin partial U$ (the boundary of $U$) and all $alpha $ (multi indice), $lim_{xto u}partial _alpha f(x)$ exist.
$endgroup$
– Surb
Dec 20 '18 at 10:06












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