Consider for some $0 < \alpha \le 1$ the space functions $x:[0,1] \to \mathbb{R}^n$ such that $x(0) = 0$ and $\sup_{s,t} \frac{\|f(t)-f(s)\|}{|t-s|^{\alpha}}$ is finite.

There are at least two reasonable norms defined on this space. The first is the Hölder norm which is just the supremum above. Another is the $1/\alpha$-variation which is the supremum over all partitions $t_0 = 0 \le t_1 \le \cdots \le t_r = 1$ of $\left(\sum_{i=0}^{r-1}|f(t_{i+1}) - f(t_i)|^{1/\alpha}\right)^\alpha$.

Let us fix $\alpha= \frac{1}{2}$ and $x(t) = \sqrt{t}$ and suppose $y:[0,1] \to \mathbb{R}$ is piecewise linear with $y(0) = 0$. It follows easily that $\lim_{t\to 0}\frac{\|x(t)-y(t)\|}{\sqrt{t}} = 1$.

This implies that there is no sequence of piecewise linear approximations to $x$ in Hölder norm.

However, it's not too hard to show that $x$ can be approximated in $2$-variation by piecewise linear functions.

My question is the following: Are piecewise linear functions dense among $1/2$-Hölder functions in the $2$-variation sense?

I'm also interested in the same question replacing piecewise linear functions by smooth functions.