For $s>1/p$ the Besov space $B_{p,q}^s([0,1])$ can be characterized in terms of the $p$-variation:
Let $p,q \in (1,\infty)$ and $s \in (0,1)$, $s>1/p$. A function $f:[0,1] \to \mathbb{R}$ is in $B_{p,q}^s([0,1])$ if, and only if, $f$ is continuous and $$\sum_{j \geq 0} \left[ 2^{j(s-1/p)} V_{p}^{(j)}(f) \right]^q < \infty \tag{1}$$ where $$V_{p}^{(j)}(f) :=\left( \sum_{k=1}^{2^j} |f(k2^{-j})-f((k-1)2^{-j})|^p \right)^{1/p}.$$
(See e.g. this paper) The assumption $s>1/p$ is important here because it guarantees the continuity of $f$.
I was wondering whether it is possible to relax the assumption $s>1/p$. More precisely, I suspect that a càdlàg (=right continuous with finite left-hand limits) function $f:[0,1] \to \mathbb{R}$ is in $B_{p,q}^s([0,1])$ if, and only if, the sum in $(1)$ is finite (no matter whether $s>1/p$ or not). Does anybody know a reference (...or, possibly, have a counterexample/an idea why one would not expect such a characterization to hold)?
