A sheaf is a presheaf with "something" added allowing us to define things locally. This task is forbidden for presheaves in general.
Specifically, a presheaf 
on a topological space 
is a sheaf if it satisfies the following conditions:
1. if 
is an open set, if 
is an open covering of 
and if 
is an element such that 
for all 
, then 
.
2. if 
is an open set, if 
is an open covering of 
and if we have elements 
for each 
, with the property that for each, 
, 
,
then there is an element 
such that 
for all 
.
The first condition implies that 
is unique.
For example, let 
be a variety over a field 
.
If 
denotes the ring of regular functions
from 
to 
then with the usual restrictions 
is a sheaf which is called the sheaf of regular functions
on 
.
In the same way, one can define the sheaf of continuous real-valued functions on any topological space, and also for differentiable functions.
