This is a question from Do Carmo's Differential Forms and Applications (question 8, chapter 2). Actually, this question was made and answered here. The problem is: The answer redirects the OP to here ($f:\mathbb{S}^1\rightarrow\mathbb{S}^1$ odd $\Rightarrow$ $\mathrm{deg}(f)$ odd (Borsuk-Ulam theorem)), but in this Differential Forms course we didn't had contact with algebraic topology whatsoever, mainly on those firsts chapters. So, there is some way, probably not so hard to prove this statement. FYI, the problem is:
Let $ F: U \subset \mathbb R ^2 \to \mathbb R^2$ be a map with $ F(x,y) = (f(x,y) , g(x,y))$, where it satisfies $F(-q)=-F(q) \quad \forall q \in D \subset U$ where $D$ is a closed disk with center the origin and set $\partial D := c$. Assume that $F$ has no zeros in $\partial D$. (a) Prove that the index $ n(F; D)$ is an odd integer. (b) Prove that $F$ has at least one zero on the disk $D$.
Is good to state that the book defines: $\displaystyle{ n(F;D) =\frac{1}{2 \pi} \int_{F \ \circ \ c} \theta _0}$ where $\displaystyle{ \theta_0 = \frac{fdg - g df}{f^2 + g^2}}$.
I know that $n(F,D)$ is the number of counterclock-wise travels around the point, in this case, the origin. I just can't use any argument. If someone could help with this, it will be very appreciated. Thanks in advance.