Is there no continuous $f:\mathbb{R}^{2^{*}}\rightarrow S^1$?

Question

This is a question from last year's exam:

Prove that there is no continuous $f:\mathbb{R}^2 -\{0\}\rightarrow S^1$ such that $f(x)=x$ on $S^1$.

Well this question is equivalent to show that $S^1$ is not a retract of $\mathbb{R}^2 - \{0\}$.

However, isn't $x\mapsto \frac{x}{|x|}$ a counterexample?

There must be a typo.. What would the original question be?

Answer 1

(Comment-turned-answer)

I agree that it must be a typo, for the example you give is exactly the example usually given to show that $\mathbb{R}^2\setminus \{0\}$ is homotopy equivalent to $S^1$.

Although it is possible that the question have many different corrections, the most likely is the following:

Prove that there is no continuous $f:\mathbb{R}^2\rightarrow S^1$ such that $f(x)=x$ on $S^1$.

This statement actually is true; there are several proofs of this fact here.