Let $f(X)=\|X\|$ and $g(X)=\|X\|^2$, $X\in \mathbb{R}^n$.

Question

Let $f(X)=\|X\|$ and $g(X)=\|X\|^2$, $X\in \mathbb{R}^n$.

Give an example of points $X$ and $Y$ at distance one unit, such that.

$a)$ $|g(Y) - g(X)|> 10^{60}$

$b)$ Show that $f$ is uniformly continuous

$c)$ Show that $g$ is not uniformly continuous

For the first clause we could consider the numbers $10^{60}$ and $10^{60}+1$. For $b)$ it is seen that it is a uniformly continuous function, but I don't know how to start with my formal proof.

Answer 1

For (b) the magic words are "triangle inequality". For (c), your example (I assume it really means vectors $(10^{60}, 0, \dotsc, 0)$ and $(10^{60}+1, 0, \dotsc, 0))$ shows you the way. All you need to show is that the function $f(x) = x^2$ is not uniformly continuous.