I'm looking for an example of some $f \in C[0,1]$ with unbounded quadratic variation in the sense of mesh converging. This definition might be most familiar in the brownian motion context. Exact definition given below
Consider a partition $$\Pi_n([0,1])=\{0=t^n_0<\cdots<t^n_{k(n)}=1\}$$ such that $\lim_{n\to \infty}\max_{i=1,\cdots,k(n)}|t^n_i-t^n_{i-1}|=0$,
$$V([0,1],\Pi_n)(f)=\sum_{i=1}^{k(n)}(f(t_{i-1})-f(t_i))^2.$$
I'm looking for some $f \in C[0,1]$ such that $$ \lim_{n \rightarrow \infty} V([0,1],\Pi_n)(f) = \infty $$
I was considering $x^{\alpha} \sin(\frac 1x)$ for $\alpha < \frac 12 $ but I'm not sure how to show this formally.