# Homework Help: Converg. Seq. of Functions, Derivatives Bounded, Limit not Differentiable

1. Nov 13, 2011

### michaelxavier

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!

Thanks!

2. Nov 13, 2011

### Dick

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