On the notion of sheaf

Question

Citing form Borceux, Handbook of categorical algebra, in the preface to volume 3:

The crucial idea behind the notion of a sheaf is to work not just with a "plain" set of elements, but with a whole system of elements at various levels. Of course, reasonable rules are imposed concerning the interactions between the various levels: an element at some level can be restricted to all lower levels and, if a compatible family of elements is given at various individual levels, it is possible to "glue" the family into an element defined at the global level covered by the individual ones. The various notions of sheafs depend on the way the words "level", "restriction" and "covering" are defined.

The easiest examples are borrowed from topology, where the various "levels" are the open subsets of a fixed space $X$: for example a continuous function on $X$ may very well be defined "at the level of the open subset $U\subseteq X$", without being the restriction of a continuous function defined on the whole of $X$.

Could someone explain me the bold part?

Answer 1

Rephrasing it, that means that you can have a continuous function $f:U \to Y$ that can't be extended continuously to the whole $X$. Moreover in that case the restriction morphism $p_{XU}$ is not surjective, of course.

Answer 2

It means you can break the domain $X$ into small pieces of open sets. A family of functions on these pieces with compatible condition determines a function on $X$. The bolded sentence means you do not need to care whether the functions on the small pieces are restricted from the functions on $X$ (but this is always the case!).