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From: Maury Barbato on 27 May 2010 00:42
I wrote:

> Hello,
> let R be the field of real numbers. Let {F_n} be
> a sequence of functions F_n:[a,b]->R such that:
>
> (I) every F_n is differentiable, and the sequence
> of the derivatives {f_n} pointwise converges to
> some continuous function f:[a,b]->R,
> (II) {F_n} pointwise converges to some
> differentiable
> function F:[a,b]->R.
>
> Can we have F'(x) =/= f(x) for some x in [a,b]?
>
> I've looked for an example, but unsuccessfully,
> so I believe the answer is negative.
>
> Thank you very much for your attention.
> My Best Regards,
> Maury Barbato

I start thinking that the answer is positive.
I made the followiing conjecture. There exists some
sequence {F_n} of differentiable functions
F_n:[0,1]->R such that:

(I) {F_n} converges uniformly to F(x) = x.

(II) the sequence of the derivatives {f_n} pointwise
converges to f(x) = 0.

What do you think about?

My Best Regards,
Maury Barbato
From: Dave L. Renfro on 27 May 2010 11:19
Maury Barbato wrote:

> I wrote:
>> Hello,
>> let R be the field of real numbers. Let {F_n} be
>> a sequence of functions F_n:[a,b]->R such that:
>
>> (I) every F_n is differentiable, and the sequence
>> of the derivatives {f_n} pointwise converges to
>> some continuous function f:[a,b]->R,
>> (II) {F_n} pointwise converges to some
>> differentiable
>> function F:[a,b]->R.
>
>> Can we have F'(x) =/= f(x) for some x in [a,b]?
>
>> I've looked for an example, but unsuccessfully,
>> so I believe the answer is negative.
>
>> Thank you very much for your attention.
>> My Best Regards,
>> Maury Barbato
>
> I start thinking that the answer is positive.
> I made the followiing conjecture. There exists some
> sequence {F_n} of differentiable functions
> F_n:[0,1]->R such that:
>
> (I) {F_n} converges uniformly to F(x) = x.
>
> (II) the sequence of the derivatives {f_n} pointwise
> converges to f(x) = 0.
>
> What do you think about?

Perhaps something in the following post will be of use:

http://groups.google.com/group/sci.math/msg/c23ad1dc3bded876

Dave L. Renfro
From: Maury Barbato on 28 May 2010 01:22
Dave L. Renfro wrote:

> Maury Barbato wrote:
>
> > I wrote:
> >> Hello,
> >> let R be the field of real numbers. Let {F_n} be
> >> a sequence of functions F_n:[a,b]->R such that:
> >
> >> (I) every F_n is differentiable, and the sequence
> >> of the derivatives {f_n} pointwise converges to
> >> some continuous function f:[a,b]->R,
> >> (II) {F_n} pointwise converges to some
> >> differentiable
> >> function F:[a,b]->R.
> >
> >> Can we have F'(x) =/= f(x) for some x in [a,b]?
> >
> >> I've looked for an example, but unsuccessfully,
> >> so I believe the answer is negative.
> >
> >> Thank you very much for your attention.
> >> My Best Regards,
> >> Maury Barbato
> >
> > I start thinking that the answer is positive.
> > I made the followiing conjecture. There exists some
> > sequence {F_n} of differentiable functions
> > F_n:[0,1]->R such that:
> >
> > (I) {F_n} converges uniformly to F(x) = x.
> >
> > (II) the sequence of the derivatives {f_n}
> pointwise
> > converges to f(x) = 0.
> >
> > What do you think about?
>
> Perhaps something in the following post will be of
> use:
>
> http://groups.google.com/group/sci.math/msg/c23ad1dc3b
> ded876
>
> Dave L. Renfro

I'm quite surprised by th fact that someone examined
similar problems before. I completely ignored Darji's
paper.
Even if it doesn't contain an answer to my question
(his results about continuous derivatives assume
that every f_n is continuous, not that f is continuous),
they are very interesting, and I'm driven to think
that my question has a negative answer.

Thank you so much, Dave, for having acquainted me
with these former investigations.
Friendly Regards,
Maury Barbato
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