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!
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See also: Prove that the function$f(x)=\sin(x^2)$ is not uniformly continuous on the domain $\mathbb{R}$. – Martin Sleziak Jul 21 '15 at 07:39
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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 $x^2$?
user2566092
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Their ratio is about $1$ but the difference will be about $2c$. And that is for ANY $c$ and $x$, as $x$ becomes large – user2566092 Feb 05 '15 at 00:49
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$x$ and $x+\delta$ are as close as you want as $x$ becomes large, but the difference between their squares is about $2c$. Just choose $x^2$ to be an arbitrarily large multiple of $2 \pi$ and $(x+\delta)^2$ to be $\pi/2$ more. – user2566092 Feb 05 '15 at 00:53
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Hint: consider the values at $\sqrt{(2 n + 1/2)\pi}$ and $\sqrt{(2n - 1/2)\pi}$
Robert Israel
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