An epsilon-delta definition is a mathematical definition in which a statement on a real function of one variable 
having, for example, the form "for all neighborhoods 
of 
there is a neighborhood 
of 
such that, whenever 
, then 
" is rephrased as "for all 
there is 
such that, whenever 
, then 
." These two statements are equivalent
formulations of the definition of the limit (
). In the second one, the neighborhood 
is replaced by the open interval 
, and the neighborhood 
by the open interval 
. For a function of 
variables, the absolute value
would be replaced by the norm 
of 
, and the open intervals by
the open balls 
and 
respectively.
This does not affect the meaning of the statement, since every neighborhood of a given point contains an open ball centered at that
point. Hence requiring that, for any 
, 
for suitable values of 
, ensures that for suitable values of 
, 
for any neighborhood 
of 
.
These suitable values of 
are, according to both versions of the definition, those belonging
to a suitable neighborhood (an open
ball in the second one).
Both statements express the fact that for all 
which lie close enough to 
, 
lies as close to 
as desired. In the second formulation this condition is
entirely expressed in terms of numbers: 
and 
are distances that measures the "closeness."
This facilitates the task of proving limits since the fundamental formulas are actually
shown by constructing, for every 
, a 
with the required property.
