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Construct an example of a sequence of functions $(f_n)$ in $BV[0, 1]$ such that $f_n \to f$ uniformly on $[0, 1]$ for some function $f \in BV[0, 1]$, whereas $(f_n)$ does not converge to $f$ in the metric $||\cdot||_{BV}$.

I was wondering if I could get a hint.

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Try a limit function which is not $BV$ (something like $\sin(1/x)$ and multiply it by smooth dampings/cutoffs near the high oscillation. – Dirk Mar 1 '13 at 12:46
Hint: $f=0$, $f_n$ becomes smaller and wigglier as $n\to\infty$. – David Moews Mar 2 '13 at 0:36

Take $f_n$ the polygonal interpolation of the points $$(0,0),(n^{-1},n^{-1}),(2n^{-1},0),\dots,(2kn^{—1},0),((2k+1)n^{-1},1).$$ Then the norm in BV of $f_n$ is approximatively $1$, and $f_n\to f$ uniformly on $[0,1]$.

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