Epsilon-Delta Definition

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.