Epsilondelta

Epsilondelta refers to the epsilon-delta definition of a limit used in real analysis and related fields. In the standard real-valued setting, one writes lim_{x→a} f(x) = L if for every ε > 0 there exists δ > 0 such that whenever 0 < |x − a| < δ, we have |f(x) − L| < ε. Here epsilon expresses an arbitrary degree of closeness of f(x) to L, and delta provides a corresponding degree of closeness of x to a that guarantees that closeness.

This formal definition provides the rigorous foundation for limits, continuity, and derivative concepts in calculus. For continuity, a related form states that lim_{x→a} f(x) = f(a), meaning that for every ε > 0 there exists δ > 0 such that if |x − a| < δ, then |f(x) − f(a)| < ε.

The epsilon-delta framework generalizes beyond real numbers to metric spaces, where absolute values are replaced by a distance function: for a limit in a metric space, one requires that d(f(x), L) < ε whenever d(x, a) < δ. In many cases, a constructive choice of δ in terms of ε (for example δ = ε/3 for a linear function) demonstrates the limit directly.

Historically, the epsilon-delta approach was formalized in the 19th century by Karl Weierstrass and later became standard in rigorous analysis, replacing earlier intuitive notions. It is closely related to but distinct from the sequential definition of limit, which uses convergent sequences; in metric spaces these notions are equivalent.

Applications include establishing limits, continuity properties, and differentiability, and serving as a precise language for analyzing behavior of functions near points of interest.