On two non-equivalent definitions of variation of a function on an interval $[a,b]$ 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!
 A: I have figured out some answers so I write them here:
Assume that $f$ is right-continuous (the case left continuous is handled similarly) and that $V^1_f[a,b] = \infty$. By definition of $V^1_f$, for any $M>0$ and $\epsilon>0$, we can find a partition $P_M = \{a = t_0 < t_1<...<t_N = b \}$ such that
$$ \sum_{i=0}^{N-1}|f(t_{i+1}) - f(t_i)| > M + \epsilon. $$
Now, since $f$ is right continuous, we can always find a number $\delta > 0$ small enough such that in any partition, e.g $P'$, with mesh lower then $\delta$, there are points $a = t'_0 <t'_1 < t'_2<...<t'_{N-1} < t'_N=b$ satisfying
$$ t_1 <t'_1<t_2<t'_2<...<t_{N-1}<t'_{N-1}, \quad and \quad \max_{i=1,...,N-1}{|t_i-t'_i|} <\xi,$$
where $\xi = \xi(\epsilon)$ is the number that makes (by the right continuity of $f$)
$$ \sum_{i=0}^{N} |f(t_i')-f(t_i)| < \frac{\epsilon}{2}.$$
Denote the variation sum of $P'$ as $V_f(P')$, by triangle inequality, we have
\begin{align}
V_f(P') &\geq \sum_{i=0}^{N-1}|f(t'_{i+1}) - f(t'_i)| \geq \sum_{i=0}^{N-1}|f(t_{i+1}) - f(t_i)| - 2\sum_{i=0}^{N-1}|f(t'_{i+1}) - f(t_i)| \\ 
&\geq \sum_{i=0}^{N-1}|f(t_{i+1}) - f(t_i)| -\epsilon > M.
\end{align}
Consequently, $V^2_f[a,b] = \infty$ (by definition above).
The case $V_f^1[a,b]< \infty$ can be treated similarly, so in short we have
$$V^1_f[a,b] = V^2_f[a,b] \quad \text{(possibly infinite) when $f$ is right or left continuous}. $$
Now, since previous statement holds for monotone functions (which have countable discontinuities), I wonder if it is true for functions that are almost everywhere (w.r.t Lebesgue measure) right-continuous (or left-continuous)?
