How to formally prove that $$f(x)= \frac{1}{x} , \quad x \in(0,1]$$ is not of bounded variation. Which partition should I try?
2026-03-25 22:02:48.1774476168
Unbounded variation $\frac{1}{x}$
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Setting $f(0)=2$ arbitrarily, we can show that $f\notin BV([0,1])$ by considering the partitions $$ \mathcal P_n =\{x_0,x_1,x_2\}= \{ 0,2^{-n},1\}.$$ Thus $$ V_0^1(f) \ge |f(x_1)-f(x_2)| =2^n-1 \to \infty. $$