If $f$ is a function of bounded variable on $[a,b]$.Then: $$V_a^b(f)=\sup_{\pi}\sum_{i=0}^n|f(x_{i+1})-f(x_i)|$$ where $\pi$ is a partition of $[a,b]$.
I wonder if$$V_a^b(f)=\lim_{\|\pi\|\to 0}\sum_{i=0}^n|f(x_{i+1})-f(x_i)|$$ if the limit exist.where $\|\pi\|$ means :$\max_{i}|x_{i+1}-x_i|$
(I got the inspiration from the definition of quadratic variation in the stochastic analysis)
No, in general the equality
$$V_a^b(f) = \lim_{\|\pi\| \to 0} \sum_{i=0}^n |f(x_{i+1})-f(x_i)|$$
does not hold. In order to see this, consider the function
$$f(x) := \begin{cases} 1 & x \neq \frac{\pi}{4} \\ 0 & x = \frac{\pi}{4} \end{cases}$$
on the interval $[a,b] := [0,1]$. Obviously, $V_0^1(f)=2$. On the other hand, if we define a sequence of partitions $\pi^n$, $n \in \mathbb{N}$, by
$$x_{i}^n := \frac{i}{n}, \qquad 0 \leq i \leq n,$$
then it is not difficult to see that
$$V^{\pi_n}(f) := \sum_{i=0}^{n-1} |f(x_{i+1}^n)-f(x_i^n)| = 0.$$