Is the function $f(x)= {\sin x \over x}$ uniformly continuous over $\mathbb{R}$? Is the function $$f(x)= {\sin x \over x}$$ Uniformly continuous over $R$
How do i approach this ? I need some hints.
 A: One way to see that $f$ is uniformly continuous (assuming the removable singularity in $0$ removed) is to note that it is differentiable, and its derivative is bounded. Then use the fact that any differentiable function with bounded derivative is uniformly continuous (even Lipschitz continuous), since by the mean value theorem we have
$$\left\lvert \frac{f(y) - f(x)}{y-x}\right\rvert = \lvert f'(\xi)\rvert \Rightarrow \lvert f(y) - f(x)\rvert \leqslant M\cdot\lvert y-x\rvert,$$
where $M$ is a bound for the derivative.
Another way to see it is to use the fact that $\lim\limits_{\lvert x\rvert\to\infty} f(x) = 0$, and that every continuous function on a compact interval is uniformly continuous. For a given $\varepsilon > 0$, choose a $K > 0$ with $\lvert x\rvert \geqslant K \Rightarrow \lvert f(x)\rvert < \varepsilon/3$. By the uniform continuity of $f$ restricted to the compact interval $[-K-1,K+1]$, there is a $\delta > 0$ such that $\lvert x\rvert, \lvert y\rvert \leqslant K+1, \lvert y-x\rvert < \delta \Rightarrow \lvert f(y)-f(x)\rvert < \varepsilon$. If $\delta$ is chosen $< 1$, that $\delta$ works then on all of $\mathbb{R}$.
A: $\mathbb{R} = (-\infty, - M ) \cup [-M,M] \cup (M,+\infty)$
Now,$\sin x \over x$ is continuous in $ [-M,M] $and hence uniformly continuous with a $\delta$ =$\delta_1$ .
Over the other intervals show that $\sin x \over x$ has a bounded derivative and hence uniformly continuous with $\delta$ = $\delta_2$ and $\delta_3$ respectively.
Then the universally $\delta$ = min ($\delta_1$,$\delta_2$,$\delta_3$)
A: Wouldn't it be easier to pick an interval, say $[-1,1]$, and say that $f(x)=\begin{cases} \frac{\sin x}x\,, & x\ne 0 \\ 1 \,, & x=0\end{cases}$ is continuous, hence uniformly continuous on $[-1,1]$, then argue that $f$ is uniformly continuous on $|x|\ge 1$, and then make a "standard" argument that allows us to glue together? (Hint: You need $\min(\delta_1,\delta_2,\delta_3)$, not just two of 'em.)
