Quotient spaces and equivalence relations

Let I(x) denote the integer part of a real number x (ie the unique integer n such that $\displaystyle n \leq x < n+1$).

I can show that $\displaystyle x$ ~ $\displaystyle y$ iff $\displaystyle I(x) = I(y)$ is an equivalence relation.

Let $\displaystyle p: {R} \rightarrow {R} /$ ~ be the quotient map, let $\displaystyle {R}/$~ be endowed with the quotient topology, and let U be an open set in $\displaystyle {R}/$~. Prove that if $\displaystyle n \in Z$ is such that $\displaystyle p(n) \in U$ then $\displaystyle p(n-1) \in U$.

Deduce that the open sets in $\displaystyle {R}/$~are the empty set, $\displaystyle {R}/$~ and the image sets $\displaystyle p(-\infty, n]$ where $\displaystyle n \in Z$.

Consider the map $\displaystyle I: {R} \rightarrow {Z}$ by $\displaystyle x \rightarrow I(x)$. Is the map I continuous (when Z is endowed with the induced topology)?