Proving that $f(x,y) = \frac{2}{x+y}$ is not uniformly continuous

Question

I'm struggling to show that $f(x,y) = \frac{2}{x+y}$ is not uniformly continuous, where the function is defined on $f:\mathbb{R}^{2}\setminus A \rightarrow \mathbb{R}$ with $A = \left \{(x,y) \in \mathbb{R}^{2} | y = -x \right \}$.

Answer 1

Look at what happens near $(0,0)$ for example on the line $y=x$.

Precisely, we want to prove the negative of the following proposition, denoting by $\mathcal{D}$ the domain of $f$

$$\forall\epsilon\gt 0,\exists\eta\gt 0,\forall (x_0,y_0)\in\mathcal{D},|(x,y)-(x_0,y_0)|\leq\eta\Rightarrow |f(x,y)-f(x_0,y_0)|\leq\epsilon$$

Let’s take $\epsilon=1$ for any $N\in\Bbb{N}*\geq 2$ we take $x_0=y_0=1/N$ and we have

$$\left|f\left(\left({1\over 2N},{1\over 2N}\right)\right)-f\left(\left({1\over N},{1\over N}\right)\right)\right|={N\over 2}\geq 1$$