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Converg. Seq. of Functions, Derivatives Bounded, Limit not Differentiable

  1. Nov 13, 2011 #1
    1. The problem statement, all variables and given/known data
    Find a sequence of differentiable functions $f_n\colon [a,b]\rightarrow\mathbb(R)$ s.t.:
    --there exists $M>0$ with $|f_n'(x)|\leq M$ for all $n\in\mathbb{N}$ and $x\in[a,b]$;
    --for all $n\in\mathbb{N}$, $|f_n(a)|\leq M$;
    --$(g_n)$ is a convergent subsequence with $lim_{n\rightarrow\ifty}g_n(x)=g$ for $f$ NOT DIFFERENTIABLE.

    2. Relevant equations
    Arzela-Ascoli for the reals.

    3. The attempt at a solution

    I have already proved, using Arzela-Ascoli, that such a $g$ exists for any sequence $(f_n)$ fulfilling the first two conditions. But I simply cannot come up with a concrete example where the limit is not differentiable!

  2. jcsd
  3. Nov 13, 2011 #2


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    f(x)=|x| is not differentiable. Can you think of a series of differentiable functions that converge to it? Hint, |x|=sqrt(x^2).
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