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

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The continuity follows from fact. The increasing behavior for follows from the Binomial theorem. For odd and , the growth behavior follows from , and the behavior in positive part. The growth behavior gives injectivity. For , we have , which shows that there is no upper bound for the image. For odd, we get that there is no lower bound for the image. Due to the Intermediate value theorem, the image is or . Hence, the power functions are surjective, and the inverse functions exist. The continuity of the inverse functions follows from fact.