variation of a function over countable intervals

Let $f$ be a function of bounded variation on $[0,1]$. Let $\{[a_n,b_n]\}_{n=1}^\infty$ such that $(a_n,b_n)$ are pairwise disjoint and $\cup_{n=1}^\infty [a_n,b_n]=[0,1]$. (for example, $[1/2, 1], [0,1/3], [1/3,1/3+1/3^2], \cdots$ ) Can we write

$$\operatorname{Var}_{[0,1]} f=\sum_{k=1}^\infty \operatorname{Var}_{[a_k,b_k]} f?$$

For your example intervals, suppose that $f$ is $0$ on $[0,1/2)$ and $1$ on $[1/2,1]$.
• I also believe this is a good counterexample. But it looks somehow fragile, because for $C^1$ functions we know that $\operatorname{Var}_{[a,b]}f = \int_a^b |f'(x)|\, dx$, and the conjecture seems to be true. Jul 22, 2014 at 7:59