
Re: Limits and Integrals
Posted:
Nov 7, 2013 2:44 AM


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.

