Epsilon-Delta Definition

An epsilon-delta definition is a mathematical definition in which a statement on a real function of one variable f having, for example, the form "for all neighborhoods U of y_0 there is a neighborhood V of x_0 such that, whenever x in V, then f(x) in U" is rephrased as "for all epsilon>0 there is delta>0 such that, whenever 0<|x-x_0|<delta, then |f(x)-y_0|<epsilon." These two statements are equivalent formulations of the definition of the limit (lim_(x->x_0)f(x)=y_0). In the second one, the neighborhood U is replaced by the open interval (y_0-epsilon,y_0+epsilon), and the neighborhood V by the open interval (x_0-delta,x_0+delta). For a function of n variables, the absolute value would be replaced by the norm ||·|| of R^n, and the open intervals by the open balls B(y_0,epsilon) and B(x_0,delta) 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 epsilon>0, f(x) in B(y_0,epsilon) for suitable values of x, ensures that for suitable values of x, f(x) in U for any neighborhood U of y_0. These suitable values of x 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 x which lie close enough to x_0, f(x) lies as close to y_0 as desired. In the second formulation this condition is entirely expressed in terms of numbers: epsilon and delta are distances that measures the "closeness." This facilitates the task of proving limits since the fundamental formulas are actually shown by constructing, for every epsilon, a delta with the required property.

See also

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.

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