
Re: Limits and Integrals
Posted:
Nov 7, 2013 12:08 PM


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.

