separated and dense in locally compact is discrete?

Question

It is true that if $A\subset \mathbb{R}$ $\delta$-separated and $\epsilon$-dense in $\mathbb{R}$ then $A$ is discrete? (This because of $\mathbb{R}$ is locally compact?

I ask this since $\mathbb{Z}$ is 1-separated and 1-dense in $\mathbb{R}$ and is discrete

Answer 1

If $\delta >0$ the any $\delta$ separated set is discrete because it has no limit points.