Positive Quadratic Variation implies infinite total variation I have just finished proving that the quadratic variation of any Brownian motion on $[0,t]$ is $t$. That is if $\mathcal{P}$ is a partition of $[0,t]$ then
$$ \lim_{\Delta t\to 0}\sum_{t_k \leq t} |B_{t_{k+1}}- B_{t_k}|^2 = t \hspace{8mm} \text{in }\hspace{4mm} L^2$$
In Stochastic Differential Equations by Oksendal, it states that if quadratic variation of a stochastic process is a.s. positive, then the total variation of the process is almost surely $\infty$. Where does this fact come from? Is the proof very intensive? It doesn't seem to appear on the wikipedia pages for https://en.wikipedia.org/wiki/P-variation, https://en.wikipedia.org/wiki/Total_variation or https://en.wikipedia.org/wiki/Quadratic_variation#Finite_variation_processes.
For my specific problem I have been able to show that $$ \mathbb{E}\left[\sum_{t_k\leq t} |B_{t_{k+1}}- B_{t_k}|\right]  = \sum_{t_k\leq t} \sqrt{\Delta t_k}$$
from properties of Brownian motion by noting that $B_{t_{k+1}}-B_{t_k}\sim \mathcal{N}(0, \Delta t_{k})$ and a a property of the expected value of the absolute value of normally distributed random variable (https://en.wikipedia.org/wiki/Normal_distribution#Moments). The right hand side can be shown to diverge, but this doesn't tell us about any of the sample paths $\sum_{t_k\leq t} |B_{t_{k+1}}(\omega)- B_{t_k}(\omega)|$.
 A: This is a general property of continuous functions.
Lemma: Suppose $f\not\equiv0$ is  continuous on $[a,b]$ and of finite variation, that is
$V_f[a,b]=\sup_{P}\sum^{n_p}_{k=1}|f(x_k)-f(x_{k-1})|<\infty$,
where supremum is taken over all partitions $P$ of $[a,b]$. Then
$$V^2_f[a,b]:=\lim_{\|P\|\rightarrow0}\sum^{n_P}_{k=1}|f(x_k)-f(x_{k-1})|^2=0$$
whereto limit is taken over all partitions $P=\{a=x_0<\ldots<x_n=b\}$ of $[a,b]$ such that $\|P\|=\max_k(x_k-x_{k-1})\rightarrow0$.
Here is a short proof:
By uniform continuity, given $\varepsilon>0$, there is $\delta>0$ such that
$$|f(x)-f(y)|<\varepsilon\quad\text{whenever}\quad|x-y|<\delta$$
For any partition $P=\{a=t_0<\ldots <t_{n_P}=b\}$ of $[a,b]$ such that $\max_{1\leq k\leq n_P}(x_k-x_{k-1})<\delta$,
$$ \begin{align}
V^2_f[a,b]&:=\sum^{n_P}_{k=1}|f(x_k)-f(x_{k-1})|^2\leq\max_{1\leq j\leq n}|f(x_j)-f(x_{j-1})|\sum^{n_P}_{k=1}|f(x_k)-f(x_{k-1})|\\
&\leq\Big(\max_{1\leq j\leq n_P}|f(x_j)-f(x_{j-1})|\Big)\,V_f[a,b]<\varepsilon\,V_f[a,b]
\end{align}
$$
Consequently
$$ V^2_f[a,b]=\lim_{\|P\|\rightarrow0}\sum^{n_P}_{k=1}|f(x_k)-f(x_{k-1})|^2=0$$

From the Lemma above, if $f$ is continuous function of finite positive quadratic variation $V^2_f[a,b]$, then $V_f[a,b]=\infty$
