TITLE: Quantile function properties
QUESTION [14 upvotes]: I am confused by "Inverse distribution function (quantile function)" section of the wikipedia page on CDFs
.  It says that $$F^{-1}(F(x)) \leq x\text{    and    }F(F^{-1}(y)) \geq y$$
However, I thought that the very definition of the inverse function was that 
$$F^{-1}(F(x)) = x\text{    and    }  F(F^{-1}(y))=y$$
Now I am starting to wonder if I've understood everything incorrectly.  Can anyone explain to me why there is an inequality instead of just an equals sign?

REPLY [4 votes]: I have uses this before, but for a visualisation try this with the CDF and its reflection

and note the vertical green line segments which strictly are not parts of the functions but appear as horizontal line segments in the reflections.