continuous and uniformly continuous I remember proving $x^3$ is not uniformly continuous on $\mathbb{R}$.
Then I read the proof of a theorem: Suppose $D$ is compact. Function $f: D \rightarrow \mathbb{R}$ is continuous on $D$ if and only if f is uniformly continuous. 
It's obvious that $f(x)=x^3$ is continuous at $x$ for all $x\in \mathbb{R}$
So does it mean if I bounded the interval $f(x)=x^3$ to, let's say, $[a,b]$ with $a<b$, then $f(x)=x^3$ is uniformly continuous?

Since your function is a polynomial and hence derivable.Thus using the fact if the derivative exists and is bounded then it is  is uniformly continuous.
$f(x)=x^3$, $f'(x)=3x^2$ and it is bounded on your interval 
$|f'(x)|\le 3b^2$ for all $x$ in $[a,b]$
