1. Note that $S=\{(\sin\theta,\cos\theta):0 < \theta < 2\pi\}$ so it is homeomorphic to the open interval $(0,\,2\pi)$. Define $f:(0,\,2\pi)\to\mathbb R$ by $f(\theta)=\tan\left(\dfrac{\theta-\pi}2\right)$.

2. Hint: Given any two points in $\mathbb R^2$, there are uncountably many continuous paths between them, and only a countable number of these paths instersect $A$.