How to think about open sets and continuous functions on discrete metrics I'm working through practice problems for the Math Subject GRE (Which seems to me to be all Analysis and Algebra even though I'd heard it was mostly multivariate calculus). This problem came up:
Let $\mathbb{Z^+}$ be the set of positive integers and let $d$ be a metric on $\mathbb{Z^+}$ be defined as follows:
$d = \begin{cases}
    1,& \text{if } m  \neq n\\
    0,& \text{if } m  =    n              
\end{cases}$
Which of the following is true:
1) For all n $\in \mathbb{Z^+}$,$\{n\}$ is open
2) Every subset of $\mathbb{Z^+}$ is closed
3) Every real valued function defined on $\mathbb{Z^+}$ is continuous.
I have a lot of trouble assessing this claims with a discrete metric and I would like some help clarifying the way I think about them. All three are true, but it's not clear to me why. More specifically:
1) Is every point of such a set an interior point? To assess this, does the open ball we draw have to be (n-1, n+1)? Does it make sense to draw a smaller open ball than this if distances in this metric are either 0 or 1? I feel like the natural language meaning of "interior point" fails here, because the natural idea of "inside" doesn't make sense. 
2) This obviously relies on one, since we test for being closed by examining if the complement is open.
3) It's not clear to me why this is true; part of that is that, as in 1), it's not clear if we can choose epsilons and deltas to be values other than 1 or 0.

$1.$ Open balls in a metric help you determine just how close two points are.  In a way, the more open balls around one that contain the other, the closer they are.  The discrete metric says all points are equidistant from each other. So, there is a smallest closed ball containing them all and smallest open ball containing only the center.
$3.$ The $\varepsilon$ in the definition of continuity can be any positive real number, including anything between $0$ and $1$.  
