Is this function defined on just the integers continuous?

Question

Define the function $f:\Bbb{Z}\rightarrow\Bbb{R}$ such that $f(z)=c$ and $c$ is any real number. Since $\forall \epsilon>0$, $\vert f(a)-f(b)\vert=\vert c-c\vert=0<\epsilon$ whenever $0<\vert a-b\vert<\delta$, for some $\delta >0$. Then $f$ must be continuous...? But the graph of this function have "breaks" in it. I'm trying to reconcile the graphical representation of continuity and its formal definition. Sorry if this is such a stupid question.

Answer 1

If $X$ and $Y$ are topological spaces then every function $f:X\to Y$ that is constant is continuous.

This because every preimage under $f$ will be an element of $\{\varnothing,X\}$ and both sets are open by definition.

Answer 2

Every function defined on the set of integers (endowed with the discrete Topology) is continuous.