How many fixed points does this function have?

Question

The function is $f :\overline{\Bbb R}\to \overline{\Bbb R}, x \mapsto x^5$.

Moreover we know that $\overline{\Bbb R}=\mathbb{R}\cup\{-\infty,+\infty\}$ is a compact set. So does it have $3$ or $5$ fixed points ?

Thanks in advance !

Answer 1

If you define $\overline{\Bbb R}=\Bbb R \cup \{-\infty \}\cup \{+\infty\}, f(-\infty)=-\infty, f(+\infty)=+\infty$ it certainly has five fixed points. You can exhibit them: $-\infty,-1,0,1,+\infty$. Once you define your terms carefully, the question has a clear answer.