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

  • #1

Homework Statement


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.

Homework Equations


Arzela-Ascoli for the reals.

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!

Thanks!
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
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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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