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Topic: Can anyone find the derivative of this?
Replies: 2   Last Post: Mar 1, 2013 7:50 AM

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Don Coppersmith

Posts: 53
Registered: 2/2/06
Re: Can anyone find the derivative of this?
Posted: Feb 27, 2013 9:56 PM
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> The function F(x) defined on the interval [0,1] as
> the sum from n=1 to n=infinity of [nx]/n!, where [.]
> denotes the greatest integer function, is strictly
> increasing and therefore has a derivative almost
> everywhere in [0,1]. The question is whether the
> derivative can be computed anywhere.
>
> Clearly, F is discontinuous at every rational number,
> but what can be said about its derivative at
> irrational numbers?
>
> I posted this question a few years ago, but thought
> in the interim someone may have some ideas or results
> that might be helpful.
>
> The range of his function is strange, containining
> only one rational number (F(0)=0). Noting that
> F(1)=e, we are led to ask: Is 0 the only algebraic
> number in the range of F?


Let x be a quadratic irrational, say x=1/sqrt(2).
Given delta, you can get a lower bound Q on denominators q
such that some fraction p/q is within delta of x:
say | p/q - x | < delta implies q > Q(delta).
This means that if a term [ny]/n! changes as y ranges through
the interval [x-delta, x+delta], we must have n > Q(delta).
That puts a bound on F(y)-F(x) for y in that interval.
I won't spoil the fun of working out the details.

Don Coppersmith



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