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Weierstrass and Jordan definitions (epsilon–delta) of continuous functions

Explicitly including the definition of the limit of a function, we obtain a self-contained definition: Given a function f: D → R {\displaystyle f:D\to \mathbb {R} } as above and an element x 0 {\displaystyle x_{0}} of the domain D {\displaystyle D}, f {\displaystyle f} is said to be continuous at the point x 0 {\displaystyle x_{0}} when the following holds: For any positive real number ε > 0, {\displaystyle \varepsilon >0,} however small, there exists some positive real number δ > 0 {\displaystyle \delta >0} such that for all x {\displaystyle x} in the domain of f {\displaystyle f} with x 0 − δ < x < x 0 + δ, {\displaystyle x_{0}-\delta <x<x_{0}+\delta,} the value of f (x) {\displaystyle f(x)} satisfies f (x 0) − ε < f (x) < f (x 0) + ε. {\displaystyle f\left(x_{0}\right)-\varepsilon <f(x)<f(x_{0})+\varepsilon.}

Alternatively written, continuity of f: D → R {\displaystyle f:D\to \mathbb {R} } at x 0 ∈ D {\displaystyle x_{0}\in D} means that for every ε > 0, {\displaystyle \varepsilon >0,} there exists a δ > 0 {\displaystyle \delta >0} such that for all x ∈ D {\displaystyle x\in D}: | x − x 0 | < δ implies | f (x) − f (x 0) | < ε. {\displaystyle \left|x-x_{0}\right|<\delta ~~{\text{ implies }}~~|f(x)-f(x_{0})|<\varepsilon.}

More intuitively, we can say that if we want to get all the f (x) {\displaystyle f(x)} values to stay in some small neighborhood around f (x 0), {\displaystyle f\left(x_{0}\right),} we need to choose a small enough neighborhood for the x {\displaystyle x} values around x 0. {\displaystyle x_{0}.} If we can do that no matter how small the f (x 0) {\displaystyle f(x_{0})} neighborhood is, then f {\displaystyle f} is continuous at x 0. {\displaystyle x_{0}.}

In modern terms, this is generalized by the definition of continuity of a function with respect to a basis for the topology, here the metric topology.

Weierstrass had required that the interval x 0 − δ < x < x 0 + δ {\displaystyle x_{0}-\delta <x<x_{0}+\delta } be entirely within the domain D {\displaystyle D}, but Jordan removed that restriction.