Here, $[\cdot ]$ denotes the greatest integer function. I tried a lot but couldn't succeed. Also, I noticed that $f$ is bounded and continuous. To prove it is not uniformly continuous on $\mathbb{R}$, we assume the contrary. Now, we let $\varepsilon = 1$. Thus, there is $\delta > 0$ such that if $x,y \in \mathbb{R}$ with $\left| x-y \right| < \delta$, then we have $\left| \sin (2\pi x [ x]) - \sin (2\pi y [ y])\right| < 1$.
I was trying to consider the values of $x$ where $f(x)=\pm 1$. But it certainly didn't help. Hints would be appreciated.
Set $\epsilon=0.5$ and let $\delta>0$.Choose a natural number $N$ such that $1/N<\delta$. Then the points $x=N$, and $y=N+\frac{1}{4N}$ satisfy $|x-y|<\delta$ and $|f(x)-f(y)|>0.5$.