Determine whether the function is uniformly continuous on the following intervals:
$f(x) = \sin (x + \frac{1}{x})$ on $(0,1)$ and $(1, \infty)$.
Here is my attempt: On $(0,1)$, choose $x_n = \pi n- \sqrt{(\pi n)^2-1}$, $y_n = \pi n + \frac{\pi}{4}- \sqrt{(\pi n +\frac{\pi}{4})^2-1}$. Then $|x_n -y_n| \rightarrow 0$, but $ |f(x_n) - f(y_n)| \rightarrow 1$, so it's not uniformly continuous on $(0,1)$. On $(1, \infty)$, $f'(x) = \cos(x+\frac{1}{x})(1-\frac{1}{x^2})$, which is bounded on $(1, \infty)$, therefore the function is uniformly continuous. Any comments are welcome.Thanks