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$f(x,y)=\left\{\begin{matrix}

e^{-1/|x-y|},x\neq y\\

0,x=y

\end{matrix}\right.$

is continuous on $\mathbb{R}^2$.

Can I conclude since $e^{-1/t}$ is continuous then for $x$,$y\in \mathbb{R}^2$, $x\neq y$,$e^{-1/|x-y|}$ is continuous on $\mathbb{R}^2$ since here $t=|x-y|$? And how to prove it when $x=y$, using the $\delta- \varepsilon$ way? Thank you a lot!