the Intersection of smooth curves can be countable but not finite set?

Question

Given two parametric $C^{1}$ curves in $R^{k}$, i.e. $\xi$ $\eta$ whose parametrization are defined on subsets $[a,b]$ and $[c,d]$ such that the ending points $\xi(b)=\eta(d)$. Assume that $\dot{\xi}\ne=0$. Can $\xi$ $\eta$ intersect on a countable set but not finite set?

Thanks.

Answer 1

Here is a simple example $(x,0)$ and $(x, x^4 \sin(\pi/x))$ for $-1 \leq x \leq 1$:

The intersections are clearly countable, by making the correspondence between integers $i$ and $1/i$.

Note that the derivative at 0 is:

${d \over d x} x^4 \sin(\pi/x) = 4 x^3 \sin \left(\frac{\pi }{x}\right)-\pi x^2 \cos \left(\frac{\pi }{x}\right)$ which is $0$ for the limit $x \to 0$.