Determine whether the following are uniformly continuous

Question

Determine whether the following are uniformly continuous

(a) $ x \rightarrow x^3, x\in \mathbb{R} $ and (b) $ x \rightarrow x^3, x\in [0,1] $.

I just don't get these, I figured that (b) is and (a) isn't but I have no idea how to prove them.

Answer 1

For $a) $, consider

$$u_n=n$$ and $$v_n=n+\frac 1 n$$ we have

$$\lim_{n\to+\infty}(v_n-u_n)=0$$ but $$\lim_{n\to+\infty}(v_n^3-u_n^3)=+\infty\ne 0$$

thus $x \mapsto x^3$ is not uniformly continuous at $\mathbb R $.

for $b) $

$$x^3-y^3=(x-y)(x^2+xy+y^2) $$ with $0\le x\le1$ and $0\le y\le 1$, thus $$|x^3-y^3|\le 3|x-y|$$

then to satisfy $|x^3-y^3|<\epsilon $, take $\eta=\frac {\epsilon}{3} $.