Determine if $h(x)=\sin(x^2)$ is uniformly continuous on $(-\infty,\infty)$

Question

Normally I know how to do these kind of questions using different kind of methods, but on this specific one I have no idea what to do:

Determine if $f(x)=\sin(x^2)$ is uniformly continuous on $(-\infty,\infty)$

any solutions or maybe advice? Thanks!

Answer 1

Hint: As $x$ becomes very large, how much of a change in $x$ does it take to complete one period of the sine function? Can you use this to answer the question? As a further hint, suppose $x$ is large and $\delta = c/x$ where $c$ is some positive constant. Then what is $(x + \delta)^2$, compared to $x^2$?

Answer 2

Hint: consider the values at $\sqrt{(2 n + 1/2)\pi}$ and $\sqrt{(2n - 1/2)\pi}$