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Topic: Limits and Integrals
Replies: 3   Last Post: Nov 7, 2013 12:08 PM

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David C. Ullrich

Posts: 3,210
Registered: 12/13/04
Re: Limits and Integrals
Posted: Nov 7, 2013 12:08 PM
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On Wed, 6 Nov 2013 23:44:02 -0800, William Elliot <marsh@panix.com>
wrote:

>On Wed, 6 Nov 2013, dullrich@sprynet.com wrote:
>

>> >If f(x,t) is uniformly continuous for x,
>
>> What does "uniformly coontinuous for x" mean?
>
>For all t, f_t(x) = f(x,t) is uniformly continuous.


In that case the answer to your question is no.

>
>> My guess is that you mean that if we say
>> f_x(t) = f(x,t) then the family {f_x} is
>> (uniformly) equicontinuous. Look up the
>> definition of "equicontinuous" and let us
>> know...

>
>Is this correct?
>
>A collection of funtions { fj:X -> Y | j in J } is equicontinous
>when for open V, there's some open U with for all j in J, x,y in X,
>(x,y in U implies fj(x), fj(y) in V.
>
>For metric spaces.
>
>A collection of funtions { fj:X -> Y | j in J } is equicontinous
>when for all r > 0 there's some s > 0 with for all j in J, x,y in X,
>(d(x,y) < s implies d(fj(x),fj(y) < r)
>

>> >does
>> >
>> >lim(x->a) integral(r,s) f(x,t) dt
>> > = integral(r,s) lim(x->a) f(x,t) dt ?

>>
>> Yes, if my guess above is correct and lim(x->a) f(x,t)
>> exists.





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