If $F$ is a function and $U$ is a set, then what does $F|U$ mean?
In http://ocw.mit.edu/courses/mathematics/18-101-analysis-ii-fall-2005/lecture-notes/lecture6.pdf, the Inverse Function Theorem, the notation is used
Also, the file in stating the Lemma 2.14 states that $A \in U$, what does this mean? A set is an element of another set? What is $A$?
EDIT: Could someone kindly explain to me what is going on in Lemma 2.16. How does $\delta$ come into play? I dont see the role in the lemma. I also don't see why this proves one-to-one