Can anybody give me an example of function which is holder continuous but is not bounded variation function?
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.
Since as sets (obviously we pick just one representative in the class of ) for and because for all (the only thing we need here is the interval to be bounded) so, since we might as well have picked 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).