Is $f(x,y) = \sin\frac{1}{xy}$ uniformly continuous on $E$, where $$ E = \lbrace(x,y) | (x-5)^2 + (y - 6)^2 < 1 \rbrace$$ ?
I tried to prove that $f$ is Lipschitz continuous, $$ \left| \sin \frac{1}{x_1y_1} - \sin \frac{1}{x_2y_2} \right| = 2\left| \sin \frac{x_2y_2 - x_1y_1}{2x_1y_1x_2y_2}\cos \frac{x_2y_2 + x_1y_1}{2x_1y_1x_2y_2} \right| \leq 2 \left| \sin \frac{x_2y_2 - x_1y_1}{2x_1y_1x_2y_2}\right| \leq \left|\frac{x_2y_2 - x_1y_1}{x_1y_1x_2y_2} \right| \leq \left|{x_2y_2 - x_1y_1} \right| $$ But I don't know what to do next, and whether I'm on the right way to study uniform continuity of $f$ .