1. ## example of function

Can anybody give me an example of function which is holder continuous but is not bounded variation function?
Thanks

2. Originally Posted by sidi
Can anybody give me an example of function which is holder continuous but is not bounded variation function?
I think that the function $x\sin\frac1x$ will satisfy both conditions. It is not of bounded variation, because it varies by an amount of approximately 1/k in each interval of the form $\Bigl[\frac1{(2k+\frac12)\pi},\frac1{(2k-\frac12)\pi}\Bigr]$. And it ought to satisfy a Hölder continuity condition with exponent 1/2 (or maybe a bit less).

I haven't checked carefully that the above example works. For an alternative construction of an example which definitely does work, look at q.2 in this pdf file.

3. Originally Posted by Opalg
For an alternative construction of an example which definitely does work, look at q.2 in this pdf file.
I have a problem with this example, I don't really understand what the function in the pdf is supposed to be like. But more importantly I have this:

Since $W^{1,p}(0,1)=C^{0,\alpha}(0,1)$ as sets (obviously we pick just one representative in the class of $u\in W^{1,p}(0,1)$) for $p> 1, \alpha = 1-\frac{1}{p}$ and because $W^{1,p}(0,1)\subset W^{1,1}(0,1)=AC(0,1)\subset BV(0,1)$ for all $p\geq 1$ (the only thing we need here is the interval to be bounded) so, since we might as well have picked $[0,1]$ instead, we would get that every Hölder continous function over a bounded interval is of bounded variation, which would contradict your examples.

I'm not sure I haven't made a mistake, but not really understanding the pdf example I can't say for sure where I made a mistake (or where he did).