Continuity of Root Function

Theorem
Let $$n \in \mathbb{N}^*$$ be a non-zero natural number.

Let $$f: \left[{0 \,. \, . \, \infty}\right) \to \mathbb{R}$$ be the real function defined by $$f \left({x}\right) = x^{1/n}$$.

Then $$f$$ is continuous at each $$\xi > 0$$ and continuous on the right at $$\xi = 0$$.

Proof
First suppose that $$\xi > 0$$.