Have I missed some information in this exercise?

Question

I'm doing an exercise about functional analysis:

We consider $E$ an Euclidean space such that $\operatorname{dim}(E)<+\infty$ and an application $P: E \rightarrow E$ such that a radius $\rho>0$ exists such that $\forall x \in \partial \mathcal{B}(0, \rho):\langle P(x), x\rangle \geq 0$. Above, $\partial \mathcal{B}(0, \rho)$ refers to the sphere of radius $\rho$.

1. Draw an illustration in 2-D.

2. We assume that $P$ does not vanish on $\overline{B(0, \rho)}$ and define $g$ as $$\forall x \in \overline{B(0, \rho)}: g(x)=-\frac{\rho}{\|P(x)\|} P(x)$$ Prove that $g$ is continuous.

3. Prove that $g$ has a fixed point.

4. Obtain a contradiction and conclude that $P$ vanishes. Explain on your picture.

In question 2, we are required to prove that $g$ is continuous. I think the continuity of $g$ depends on the continuity of $P$. I'm not sure if the fact that $P$ is called an application has some mathematical meaning in this context. Maybe I'm wrong, but I think I can not prove that $g$ is continuous with the existing information.

Could you please shed me some light on this issue?