I am wondering about the following, possibly naive, question.
Suppose I have a smooth curve, which intersects the horizontal axis in a manner that leads to an accumulation point. More precisely, given a $C^2$ curve $y=f(x)$ such that the set $\{x$ s.t. $f(x)=0\}$ is a set of measure zero (for the Lebesgue measure) which contains an accumulation point, say $x=0$. Does it mean that the $\mathcal{H}^1$ Hausdorff measure of the curve (as a set in $\mathbb{R}^2$) is infinite?
(I hope that it is the case, but I would be very happy with a counter-example as well).
No, it does not have to be infinite. For example, the graph of $y=x^{3}\sin (1/x)$, $-1\le x\le 1$, is $C^2$ smooth and its length is finite.
Actually, every $C^2$ curve with compact parameter domain has finite length.