For any x ∈ R define the floor of x, denoted [x],to be the largest integer y with y ≤ x. Then define a function f : R → R by f(x) =[x]. Show that f is continuous at all c ∈ R \ Z and discontinuous at all c ∈ Z.
Not sure how to even start the question. How do I even define the floor of x? (Probably basic but I am stuck).
It looks obvious. For $x$ not an integer, $[x] < x < [x]+1$, let $\epsilon =min (\frac{x-[x]}{2},\frac{1+[x]-x}{2})\gt 0$, then for $(x-\epsilon \lt y \lt x+\epsilon)$, $f(x)=[x]$ a constant. Therefore $f(x)$ is continuous. For $x$ an integer, for any $\epsilon \gt 0$, $ [x-\epsilon]=x-1$, while $[x+\epsilon]=x$, so not continuous at $x$.