While studying stochastic process, I encountered two non-equivalent definitions of variation as follows:
Definition 1: Let $f$ be a function on $[a,b]$, the variation of $f$ is defined as
$$V^1_f[a,b] = \sup_{P} \sum^{n-1}_{i=0} |f(t_{i+1}) -f(t_i)|,$$
where the supremum is taking over all partitions $a = t_0 <t_1<...< t_n =b$ of $[a,b]$.
Definition 2: Let $f$ be a function on $[a,b]$, the variation of $f$ is defined as
$$ V^2_f[a,b] = \lim_{\|P\| \rightarrow 0} \sum^{n-1}_{i=0} |f(t_{i+1}) -f(t_i)|, $$
where $\| P \|$ is the mesh of the partition $a = t_0 <t_1<...< t_n =b$ of $[a,b]$, i.e, $\max_i\{t_{i+1} - t_i\}$.
The second definition means $V_f^2[a,b]$ is the finite number (if exists) such that, given $\epsilon >0$, $\exists \delta >0$ whenever $\|P\| < \delta$ then $\Big|\sum^{n-1}_{i=0} |f(t_{i+1}) -f(t_i)| - V_f^2[a,b]\Big| <\epsilon $. To see why they are not equivalent, just take $f$ to be the Dirichlet function on $[0,1]$.
In this post, I'm looking for sufficient conditions for the equality $V_f^1 = V_f^2$, I couldn't find much information on the internet so I hope to get your help here. Two well-known sufficient conditions are 1. $V^1_f[a,b] <\infty$ plus $f$ is continuous and 2. $f$ is monotone, which are proved in many texts. But I would like to see other conditions rather than just continuity of $f$. For example, are they still equal if $f$ is just right-continuous, or left-continuous (possibly just almost everywhere), etc.?
Another question is when $f$ is continuous but $V_f^1[a,b] = \infty$, is it true that $V_f^2[a,b] = \infty$? The latter means that given a number $M>0$, then there is $\delta$ such that whenever $\|P\| < \delta$ we have $\sum^{n-1}_{i=0} |f(t_{i+1}) -f(t_i)| > M$.
For example, a Brownian motion $B_t$ is continuous but $V^1_B[a,b] = \infty$, I wonder if $V_B^2[a,b]= \infty$?
Thank you for your help!