Re: [分析] f(x,y) continuous

[GSXSP (Gloria)]

※ 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.


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→ kerwinhui   : any condition on A?  E.g., false for A=R.
→ GSXSP       : Can you give me an example for A=R, thanks.
→ r19891011   : Int_{y \in R}exp(-ixy) dy=2 $pi $delta(x)
Add a constraint f(x,y) \in R, real function

→ willydp     : f(x,y)=(x/√π)exp(-y^2/x^2)
→ willydp     : x∈[-1, 1], y∈R
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

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→ kerwinhui   : f(x,y)=x for all (x,y); then int f(x,y) dy does
→ kerwinhui   : not give a finite result for x!= 0, and 0 at x=0
→ kerwinhui   : you can introduce a cut-off function in y to get
→ kerwinhui   : e.g. f(x,y)=x eta(xy), eta = a bump function
→ njru81l     : f(x,y)=x eta(xy) is an unbounded functon.
→ njru81l     : If c satisfies eta(c)=/=0,then f(x,c/x)=x eta(c)
→ njru81l     : |f(x,c/x)|→∞ as x→∞
→ kerwinhui   : No, eta is a bump function, i.e., eta supported
→ kerwinhui   : on [-1-epsilon, 1+epsilon] and eta(x)=1 for all
→ kerwinhui   : x in [-1,1]