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Given a function that is absolutely continuous on all of $\mathbb{R}$ (not some closed and bounded interval such as $[a,b]$ for a<b) is not necessarily of bounded variation on all of $\mathbb{R}$. Is there an example of a function for which this is true? I am trying to look for one but cannot seem to find it.

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    $\begingroup$ Isn't $f(x)=x$ an example? $\endgroup$
    – saulspatz
    Commented Apr 25, 2021 at 21:55

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The identity function $f(x)=x$: it's of bounded variation on any bounded set but not on all of $\mathbb{R}$

Or if you want it to be the integral of its derivative on every $[-\infty, a]$, $f(x)=\exp(x)/(1+\exp(x))$

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