Re: [分析] f(x,y) continuous

看板Math作者 (Gloria)時間11年前 (2014/11/06 10:11), 11年前編輯推噓0(0推 0噓 10→)
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※ 引述《GSXSP (Gloria)》之銘言： Prove or disprove that If f(x,y) bounded and continuous in (x,y), then Int_{y \in A } f(x,y) dy is continuous in x. -- ※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 166.170.48.149 ※ 文章網址: http://www.ptt.cc/bbs/Math/M.1414787482.A.40C.html
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11/01 07:45,
any condition on A? E.g., false for A=R.
11/01 07:45
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11/01 08:12,
Can you give me an example for A=R, thanks.
11/01 08:12
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11/02 02:12,
Int_{y \in R}exp(-ixy) dy=2 $pi $delta(x)
11/02 02:12
Add a constraint f(x,y) \in R, real function
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11/02 07:10,
f(x,y)=(x/√π)exp(-y^2/x^2)
11/02 07:10
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11/02 07:12,
x∈[-1, 1], y∈R
11/02 07:12
int_{y\in R} (x/√π)exp(-y^2/x^2) dy = x^2 is continuous did I miss something? ※ 編輯: GSXSP (132.239.223.126), 11/06/2014 02:00:51 -- ※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 132.239.223.126 ※ 文章網址: http://www.ptt.cc/bbs/Math/M.1415239895.A.1B5.html
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11/06 23:12, , 1F
f(x,y)=x for all (x,y); then int f(x,y) dy does
11/06 23:12, 1F
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11/06 23:13, , 2F
not give a finite result for x!= 0, and 0 at x=0
11/06 23:13, 2F
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11/06 23:13, , 3F
you can introduce a cut-off function in y to get
11/06 23:13, 3F
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11/06 23:17, , 4F
e.g. f(x,y)=x eta(xy), eta = a bump function
11/06 23:17, 4F
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11/07 00:00, , 5F
f(x,y)=x eta(xy) is an unbounded functon.
11/07 00:00, 5F
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11/07 00:02, , 6F
If c satisfies eta(c)=/=0,then f(x,c/x)=x eta(c)
11/07 00:02, 6F
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11/07 00:03, , 7F
|f(x,c/x)|→∞ as x→∞
11/07 00:03, 7F
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11/09 13:05, , 8F
No, eta is a bump function, i.e., eta supported
11/09 13:05, 8F
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11/09 13:06, , 9F
on [-1-epsilon, 1+epsilon] and eta(x)=1 for all
11/09 13:06, 9F
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11/09 13:06, , 10F
x in [-1,1]
11/09 13:06, 10F
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文章代碼(AID): #1KMjZN6r (Math)