  A connected compact subset $E$ of $\mathbb{R}^N$ is said to be a strict
Whitney set if there exists a real-valued $C^1$ function $f$ on $\mathbb{R}^N$
with $\nabla f|_E\equiv 0$ such that $f$ is constant on no non-empty relatively
open subsets of $E$. We prove that each self-similar arc of Hausdorff dimension
$s>1$ in $\mathbb{R}^N$ is a strict Whitney set with criticality $s$. We also
study a special kind of self-similar arcs, which we call "regular" self-similar
arcs. We obtain necessary and sufficient conditions for a regular self-similar
arc $\Lambda$ to be a $t$-quasi-arc, and for the Hausdorff measure function on
$\Lambda$ to be a strict Whitney function. We prove that if a regular
self-similar arc has "minimal corner angle" $\theta_{\min}>0$, then it is a
1-quasi-arc and hence its Hausdorff measure function is a strict Whitney
function. We provide an example of a one-parameter family of regular
self-similar arcs with various features. For some value of the parameter
$\tau$, the Hausdorff measure function of the self-similar arc is a strict
Whitney function on the arc, and hence the self-similar arc is an
$s$-quasi-arc, where $s$ is the Hausdorff dimension of the arc. For each
$t_0\ge 1$, there is a value of $\tau$ such that the corresponding self-similar
arc is a $t$-quasi-arc for each $t>t_0$, but it is not a $t_0$-quasi-arc. For
each $t_0>1$, there is a value of $\tau$ such that the corresponding
self-similar arc is a $t_0$-quasi-arc, but it is a $t$-quasi-arc for no $t\in
[1, t_0)$.
