Consider for some $0 < \alpha \leq 1$ 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.



Are piecewise linear curves dense among Hölder curves?
To expand on fedja's comment :


For $p>1$, a function $f$ on $[0,1]$ is in the $p$-variation closure of smooth functions $C^{0,p-var}$ iff


$$\lim\_{\delta \rightarrow 0}\;\;\; \sup\_{\substack{0=t\_0<\ldots< t\_m=1 \\\ |t\_{i+1}-t\_i|\leq\delta}} \sum (f(t\_{i+1})-f(t\_i))^p = 0. \label{rel}$$ 


Then the function $g(x) = \sum\_{i \geq 1} c^{-i/p} \sin(c^i x)$ is $(1/p)$-Hölder, but does not satisfy this relation (for $c$ large enough).


Note that any continuous function of finite $q$-variation, for some $q< p$ (such as your square root example), is in $C^{0,p-var}$.


For $p=1$, $C^{0,1-var}$ is the space of absolutely continuous functions.


You can find these results e.g. in Subsection 5.3.3 of Friz&Victoir "Multidimensional stochastic processes as rough paths" ([pdf](http://page.math.tu-berlin.de/~friz/master4_May6th.pdf "example"))


