It is known that if $f(x):\mathbb{R}\to \mathbb{R}$ is absolutely continuous on $[a,b]$, then it has bounded variation on $[a,b]$.
Is it also true for uniformly continuous functions? And if not then what is an example?
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1Wikipedia gives the standard example $f(x) = x \sin (1/x)$ extended by continuity on $[0,2/ \pi]$. – Crostul May 12 '17 at 17:34
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2For an extreme example, take your favorite pathological continuous function on a compact interval, say a nowhere differentiable continuous function. (If you want an example that's uniformly continuous on the entire real line, then extend the function to be constant outside the interval.) – Dave L. Renfro May 12 '17 at 18:17
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You can take $f(x)=x$, it is uniformly continuous but does not have bounded variation over $\mathbb{R}$
Even you can find uniformly continuous functions without bounded variation in an interval say $[0,1]$.
You can make such a function by joining discrete line segments. Consider any continuous function passing through the points $(1/2n,1/n)$ and $(1/(2n+1),0)$, this is composed of linear segments. It must have infinite variation because its variation is the summation of the harmonic series which diverges.
