Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2007 / 4 / c

Because the image is integer-valued, the open interval $(n-\frac{1}{2},n+\frac{1}{2})$ contains exactly one possible image value, namely $n$. ThusThe right side is open in $A$ by continuity, soThis includes the possibility of an empty fiber. The source notation $f^{-1}(n)$ is understood as the preimage of the singleton $\{n\}$.