 TOPICS  # 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.

Continuous Function, Epsilon, Epsilon-Delta Proof, Limit

This entry contributed by Margherita Barile

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Barile, Margherita. "Epsilon-Delta Definition." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Epsilon-DeltaDefinition.html