Show that none of the functions $f(x)=\sqrt[n]{x}$, for $n\in \Bbb N$, $n\ge2$ are Lipschitz continuous.
So i did it this way: $|\sqrt[n]{x}-\sqrt[n]{y}|=|x-y||\frac{1}{x^{\frac{n-1}{n}}+x^{\frac{n-2}{n}}y^{\frac{1}{n}}+...+x^{\frac{1}{n}}y^{\frac{n-2}{n}}+y^{\frac{n-1}{n}}}| \le (?)L|x-y|$ and argued that for very small x and y (close to 0) the denominator is very small, therefore the whole expressions is big, and the whole fraction cannot be limited by a real number (L), so these functions are not Lipschitz continuous.
Am I corrected? Is there any nicer, more formal way to show it?