# Proof

(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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### References

#### Bibliography

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