Prove uniform continuity or discontinuity of function $\sqrt{x}\sin{x}$ in $(0,\infty)$

Question

I am given a function as is mentioned in title. I want to prove it uniformly continuous or discontinuous. But I am not able to do it. Any help would be appreciated.

Answer 1

Let $$x_n = 2n^2\pi + \frac{1}{n} \quad \quad y_n = 2n^2\pi$$ Then with $f(x) = \sqrt{x}\sin x$, we have $$|f(x_n)-f(y_n)| = \sqrt{2n^2\pi+\frac{1}{n}}\sin\frac{1}{n}\to \sqrt{2\pi} \quad \quad \text{ as } \quad n\to\infty$$ Hence it is not uniformly continuous over $(0,\infty)$.