I have to prove that $$ \begin{cases} \begin{array}{cc} \sin(1/x) & x\in \Big(0,\frac{2}{\pi}\Big] \\ 0 & x=0 \end{array} \end{cases} $$ is not of bounded variation. I am unable to figure out the partition of this, so please provide me any partition s.t. it is not of BV.
2026-03-27 20:01:11.1774641671
$\sin(1/x)$ is not BV.
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Hint:
Consider $x_k = [\pi(k + 1/2)]^{-1}$ for $k = 0, 1 \ldots, n$
$$\sum_{k=1}^n \left| \sin(1/x_k) - \sin(1/x_{k-1})\right| = \cdots$$