winding function and trigonometric functions.

Question

here are the questions:

I am stuck in (7), I know that the preimage of an open arc is an interval, but then what ? how will this prove that $W$ is continuous?

for 8(a) and 8(b), I do not know at all what should I do? any help will be appreciated.

$\mathcal{F}$ is the set of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$.

Answer 1

The preimage of an open set being open is the topological definition of continuity. For 8 consider $f(t)=(cos(t),sin(t))$