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

Proof

The continuity follows from fact. The increasing behavior for ${}x\geq 0$ follows from the Binomial theorem. For ${}n$ odd and ${}x<0$, the growth behavior follows from ${}x^{n}=-(-x)^{n}$, and the behavior in positive part. The growth behavior gives injectivity. For ${}x\geq 1$, we have ${}x^{n}\geq x$, which shows that there is no upper bound for the image. For ${}n$ odd, we get that there is no lower bound for the image. Due to the Intermediate value theorem, the image is ${}\mathbb {R}$ or ${}\mathbb {R} _{\geq 0}$. Hence, the power functions are surjective, and the inverse functions exist. The continuity of the inverse functions follows from fact.

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