$\newcommand\P{\mathcal P}$A "partition" $P$ (of the interval $[0,1]$) is a finite sequence $(t\_0,\dots,t\_n)$ such that $0=t\_0<\cdots<t\_n=1$; then the mesh of $P$ is $\|P\|:=\max\_{1\le j\le n}(t\_j-t\_{j-1})$.


Fix any sequence $\P:=(P\_k)$ of "partitions" $P\_k=(t\_{k,0},\dots,t\_{k,n\_k})$ such that $\|P\_k\|\to0$ (as $k\to\infty$).


For a real-valued function $f$ on $[0,1]$, define its quadratic variation (with respect to $\P$) by the formula
$$[f]\_\P:=\limsup\_{k\to\infty}\sum\_{j=1}^{n\_k}(f(t\_{k,j})-f(t\_{k,j-1}))^2.$$


Suppose now that $f$ is differentiable and $[f]\_\P<\infty$. Does it then necessarily follow that
$[f]\_\P=0$?


This question is a modification of the (now answered, positively) previous question [A dichotomy for the quadratic variation of differentiable functions?](https://mathoverflow.net/q/397707/36721). The difference is that now the sequence $\mathcal P$ is fixed.



A narrower dichotomy for the quadratic variation of differentiable functions?
Using your previous example of $f(x) = x^2 \cos(x^{-4})$. Note that on any interval $[\epsilon,1]$ the function is continuously differentiable, and hence has quadratic variation 0.


Construct $P\_k$ so that the following points are contained in the partition:


1. 0
2. $\frac{1}{\sqrt[4]{\pi}} \ell^{-1/4}$ for natural $\ell$ between $k$ and $2k$.
3. choose a partition of $[(k\pi)^{-1/4}, 1]$ such that the quadratic variation of $f$ on that interval is less than 1/10, and such that the points are no further than $1/k^{1/4}$ apart.


With this the sum relative to $P\_k$ of
$$ \sum (f(t\_{k,n}) - f(t\_{k,n-1}))^2 $$
evaluates to approximately $\ln(2)$.


