In the link http://en.wikipedia.org/wiki/Open_and_closed_maps,
it says
"To every point on the unit circle we can associate the angle of the positive x-axis with the ray connecting the point with the origin. This function from the unit circle to the half-open interval $[0,2π)$ is bijective, open, and closed, but not continuous."
So is $(1,0)$ the only point on the circle that isn't continuous?
Also, the Wikipedia article says the floor function is both open and closed. I don't see how it can be open since the image of the set $(0,1)$ under the floor function is $\{0\}$, which is definitely not open.
Best Answer
1) Yes, $(1,0)$ is the only point on the circle where this function is discontinuous.
2) You left out something: it says "the floor function from R to Z is open and closed, but not continuous." $\{0\}$ is open as a subset of the discrete space $\mathbb Z$.