Define uniform continuity for functions defined on a (bounded or unbounded) interval in $\mathbb{R}$.

Is it true that a real function defined and uniformly continuous on $[0,1]$ is necessarily bounded?

Is it true that a real function defined and with a bounded derivative on $[1, \infty)$ is necessarily uniformly continuous there?

Which of the following functions are uniformly continuous on $[1, \infty)$ :

(i) $x^{2}$;

(ii) $\sin \left(x^{2}\right)$;

(iii) $\frac{\sin x}{x}$ ?

Justify your answers.