$\frac1{x ^ 2 + y^2}$ is uniformly continuous in your domain?

Question

They let me see if the function$$\frac1{x ^ 2 + y^2}$$ is uniformly continuous in their domain but I have not been able to solve the problem, will anyone have any suggestions on how to solve the problem?

Answer 1

Assuming the domain is $\mathbb{R}^2\setminus \{(0,0)\}$, the answer is no.

Let $\varepsilon > 0$ and consider the points $(x, 0)$ and $(x+\varepsilon, 0)$ for some $x > 0$. We have $$\left|\frac1{x^2} - \frac1{(x+\varepsilon)^2}\right| = \frac{2x\varepsilon + \varepsilon^2}{(x+\varepsilon)^2x^2}\xrightarrow{x\to 0} \infty$$ but $$\|(x+\varepsilon, 0) - (x, 0)\| = \varepsilon$$ We conclude that $(x,y) \mapsto \frac1{x^2+y^2}$ is not uniformly continuous.