Real numbers/kth root of real number/Intermediate value theorem/Fact

Let ${}n\in \mathbb {N} _{+}$. For ${}n$ odd,

the power function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{n},$

is continuous, strictly increasing, bijective, and the inverse function

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{1/n},$

is strictly increasing and continuous.

For ${}n$ even, the power function

$\mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} _{\geq 0},x\longmapsto x^{n},$

is continuous strictly increasing, bijective, and the inverse function

$\mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} _{\geq 0},x\longmapsto x^{1/n},$

is strictly increasing and continuous.

Proof, Write another proof