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

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William Elliot

Posts: 1,437
Registered: 1/8/12
Re: Limits and Integrals
Posted: Nov 7, 2013 2:44 AM
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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.

> 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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