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Why is $x^{1/n}$ continuous for positive $x,n$ where $n$ is an integer? I can't see how it follows from the definition of limit. And I don't see any suitable inequalities so is this an application of Bernoulli's or Jensen's inequality?

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Is $n$ a positive integer, a positive rational, or a positive real number? –  Arturo Magidin Sep 23 '11 at 18:47
    
$n$ is a positive integer. –  student Sep 23 '11 at 18:48

2 Answers 2

up vote 8 down vote accepted

The idea is that $f(x) = x^{1/n}$ is Lipschitz in a small neighborhood of any point $x > 0$.

We need the following estimate. For $x, y > 0$, denote $u := f(x)$ and $v := f(y)$, so that $x = u^n$ and $y = v^n$. Then we can bound $|f(x)-f(y)|$ by $$ \begin{align*} |f(x) - f(y)| = |u-v| &= \left| \frac{x-y}{u^{n-1} + u^{n-2} v + \cdots + u v^{n-2} + v^{n-1}} \right| \\ &\leqslant \frac{|x-y|}{u^{n-1}} = \frac{|x-y|}{x^{\frac{n-1}{n}}}. \end{align*} $$ Therefore, for any $a > 0$, $f$ is $\ell$-Lipschitz in the neighborhood $\left[\frac{a}{2}, 2a \right]$ of $a$ for $\ell = \ell(a) = \left( \frac{a}{2} \right)^{- \frac{n-1}{n}}$. Continuity at $a$ follows from this. $\qquad \diamond$


Remarks.

  1. In fact, $f$ is also differentiable at every $a > 0$. Moreover, it is possible to strengthen the above proof to obtain the derivative as well.

  2. It turns out the function is Hölder continuous on every interval $[0, a]$, though it is not Lipschitz (assuming $n > 1$). Therefore, $f$ is continuous but not differentiable at the origin.

  3. While Mariano's answer explicitly uses that $f(x)$ is the inverse function to $g(x) = x^n$, we also exploit this fact -- but implicitly. Indeed, since $g$ is invertible and has nonzero derivative at every point in $[0, \infty)$, the inverse function theorem guarantees that $f$ is also differentiable at every $a > 0$.

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@student I haven't shown the full proof, of course, since it seemed that you just wanted an inequality that would aid the usual $\epsilon$-$\delta$ proof. The remaining parts of the proof are fairly standard. But if you want further hints, I can update my answer. –  Srivatsan Sep 23 '11 at 18:54
    
Sketch is fine. I understood it and found details. TY. –  student Sep 23 '11 at 19:43
    
@student, I have expanded my answer giving more pointers to standard concepts related to continuity. I hope you find this helpful. –  Srivatsan Jan 2 '12 at 16:03

One way to show this is to observe that it is the inverse function of $x\mapsto x^n$, which is itself continuous.

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True but I was asked to show a function which is sum of a polynomial and nth roots to be a continuous using epsilon-delta technique so Srivatsan Narayanan's answer seems better for my purpose. I'm not sure how to apply epsilon-delta with inverse functions. –  student Sep 24 '11 at 17:16

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