Are there continuous functions having infinite $p$-variation for any $p\ge 1$?
I guess yes since for any $p=1,2,\dots$ and any $x\in \mathbb R$, there presumably is a function $f : [p-1,p] \rightarrow \mathbb R$ with $f(p-1)=x$ having infinite $p$-variation. Concatenation yields the required function.
Two questions arise from that:
- What are concrete example for functions having infinite $p$-variation? For $p\in[1,2]$, I know that paths of Brownian motion do the job. But is there a natural example for general $p$?
- Are there continuous function which have locally infinite $p$-variation, in the sense that on any non-empty interval the $p$-variation is infinite for any $p\ge 1$?
