TITLE: Why is the real function $x^{1/3}$ continuous?
QUESTION [5 upvotes]: I want to prove that the function $f(x) := x^3$ for all real $x$ defines a homeomorphism from $\mathbb{R}$ to $\mathbb{R}$. But I am finding it difficult to prove that the inverse map is continuous!

REPLY [4 votes]: HINT: As far as I know you could use the very powerful tool that any continuous bijection on a compact topological space onto a topological  Hausdorff space is a homeomorphism.