$g(x)=sin(x)$ and $h(x)=sin(x^2)$ uniformly continuous on $(-\infty, \infty)$?

Question

determine whether $g(x)=\sin{(x)}$ and $h(x)=\sin{(x^2)}$ is uniformly continuous on $(- \infty, + \infty)$?

basically I have no clue, trying to learn this uniform continuity stuff, so anyone got any solutions/tips on this one? thanks :)

Answer 1

For $g(x)=sinx$, use the fact that $|g'(x)| \leq1$ ; then, for $x,y$ , you have $|g(x)-g(y)|\leq |g'(x)(x-y)|\leq |x-y|$ (using MVThm here). Now, just let |x-y| become as small as you want.

EDIT: Corrected mistake, deleted second part of answer.