Can anyone give an example of a function $f: \mathbb{R} \rightarrow \mathbb{R}$, $f \in C_b$ or some other "normal" bounded function space where the variation is unbounded on some finite interval?
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Of course. You will just need to make the function zigzagged enough. The example provided in the answer below $\sin 1/x$ is an obvious example. Besides this, the famous Weierstrass function also furnishes an example, since it is nowhere differentiable (BV funcs are a.e. differentiable). – Vim Nov 28 '19 at 09:45
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$f(x)=\sin (\frac 1 x)$ for $0<x \leq 1$, $f(0)=0$ is a standard example.
Kavi Rama Murthy
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