Let $f: [0,\infty)\to \mathbb{R}$. The quadratic variation of $f$, if it exists, is defined as the function $\langle f\rangle: [0,\infty) \to \mathbb{R}$ with $$ \langle f\rangle_t := \lim_{n\to \infty} \sum_{t_i \in \pi_n(t)} \left( f(t_{i+1}) - f(t_i) \right)^2 $$ for $t \in [0,\infty)$ where $\{\pi_n(t): n\in \mathbb{N}\}$ is a sequence of refining partitions of $[0,t]$.
I am looking for an example of a function $f$ such that $f$ is continuous and its quadratic variation $\langle f\rangle$ exists, but $\langle f \rangle$ is not continuous.
Motivation: In probability lecture notes, one sometimes reads (e.g. in the context of the pathwise Ito formula) that a path $X(\omega)$ is assumed to be continuous with continuous quadratic variation. Therefore, I would like to understand why it is necessary to explicitly demand the quadratic variation to be continuous if this is desired.
An example is given in Appendix 5.2 of Coquet, Jakubowski, Mémin, and Słominski, Natural decomposition of processes and weak Dirichlet processes, pp. 81–116, Springer, Berlin, Heidelberg, 2006.
For completeness, I replicate the example below: we restrict ourselves to the time interval $[0,1]$. Let $f \in C[0,1]$ be defined by $f(t) = 0$ when $t = 1 - 2^{1-2p}$ and $f(t) = \frac{1}{p}$ when $t = 1 - 2^{-2p}$, where $p \in \mathbb{Z}^+$. We complete the construction of $f$ by linearly interpolating between these points. The graph of $f$ looks like a sequence of shrinking scalene triangles as you move forward in time.
By construction, is is clear that $f$ is of bounded variation on all intervals $[0,t]$ with $t < 1$; hence its quadratic variation on those intervals vanish.
By considering the quadratic variation on $[0,1]$ along the sample points $T = \{ 1 - 2^{-2k} \, : k \geq 1 \}$, you get that this quantity is infinite.
Finally, you may construct a sequence of refining partitions of $[0,1]$, say $(\pi_n)_{n=1}^\infty$, given by:
\begin{align*} \pi_n = \bigcup_{j \leq 2^{2n} - 1} \{j 2^{-2n}\} \cup \bigcup_{k \geq n} \{ 1 - 2^{-2k} \} \end{align*}
You can show that along this sequence, the pathwise quadratic variation takes up its entire mass at $t=1$.