Let $ f (X) = || X || $ and $ g (X) = || X || ^ 2 $, $ X \in \mathbb{R} ^ n $.
a) Give an example of points $ X $ and $ Y $ at a distance of one unit, such that
\begin{equation} | g (Y) - g (X) |> 10 60 \end{equation}
b) Show that $ f $ is uniformly continuous.
c) Show that $ g $ is not uniformly continuous.
Uniform continuity. Let $S\subset \mathbb{R}^n$. A function $: S → \mathbb{R}^m$ is uniformly continuous over $S$ if for all tolerance $\varepsilon>0$, there exists a precision $> 0$ such that if $X$ and $Z∈S$ at a distance less than , then $F(X)$ and $F(Y)$ are at a distance less than $$. This is if $||X-Z||< $ then $‖F(X)-F(Z)‖ <\varepsilon$.
I was trying to make a draft, but I don't know if this is valid: $|| f (X) -f (Z) || = ||\: || X || - || Z ||\: || \leq ||\: ||X||\: || - ||\: ||Z||\: ||$.