(related to Proposition: Nth Roots of Positive Numbers)

By hypothesis, \(n\) is an integer with \(|n|\ge 2\) and \(\mathbb R_{+}\) denotes the set of positive real numbers, including \(0\).

Note that the \(n\)-th power function \(f:x\to x^n\) means a multiplication of the continuous function \(id:x\to x\) with each other (\(n\) times). Because arithmetic operations (like multiplication of continuous functions with each other) preserve the continuity, the \(n\)-th power is a continuous function. Moreover, for all \(x\ge 0\) it follows from \(y > x\) that \(y^n > x^n\), if \(n\ge 2\), and \(y^n < x^n\), if \(-n\le -2\). Thus, the \(n\)-th power is strictly monotonically increasing, if \(n\ge 2\) and strictly monotonically decreasing, if \(-n\le -2\) on any closed real interval \([0,y]\) for every \(y > 0\).

For these reasons, the \(n\)-th power is invertible on whole \(\mathbb R_{ + }\) and the inverse function \(g:x\to\sqrt[n]{x}\) exists, is continuous and strictly monotonically increasing (respectively decreasing) on whole \(\mathbb R_{ + }\).

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983