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Whenever a point x is within δ units of p, f(x) is within ε units of L

In calculus, the (ε, δ)-definition of limit ("epsilon-delta definition of limit") is a formalization of the notion of limit. A form of it with the (ε, δ) notation (and exploiting the real inequalities as below) was defined by 19th-century French mathematician Augustin Louis Cauchy,[1][2] and it was later formalized and made logically rigorous in the 19th-century by the German mathematician Karl Weierstrass.

Contents

Informal statement

Let ƒ be a function. To say that

 \lim_{x \to c}f(x) = L \,

means that ƒ(x) can be made as close as desired to L by making x close enough, but not equal, to c.

How close is "close enough to c" depends on how close one wants to make ƒ(x) to L. It also of course depends on which function ƒ is and on which number c is. The positive number ε (epsilon) is how close one wants to make ƒ(x) to L; one wants the distance to be no more than ε. The positive number δ is how close one will make x to c; if the distance from x to c is less than δ (but not zero), then the distance from ƒ(x) to L will be less than ε. Thus δ depends on ε. The limit statement means that no matter how small ε is made, δ can be made small enough.

The letters ε and δ can be understood as "error" and "distance", and in fact Cauchy used ε as an abbreviation for "error" in some of his work.[1] In these terms, the error (ε) can be made as small as desired by reducing the distance (δ).

This definition also works for functions with more than one input value. In those cases, δ can be understood as the radius of a circle or sphere or higher-dimensional analogy, in the domain of the function and centered at the point where the existence of a limit is being proven, for which every point inside produces a function value less than ε from the value of the function at the limit point.

Precise statement

The (ε, δ)-definition of the limit of a function is as follows:

Let ƒ be a function defined on an open interval containing c (except possibly at c) and let L be a real number. Then the formula

 \lim_{x \to c}f(x) = L \,

means

for each real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − c| <  δ, we have |ƒ(x) − L| < ε,

or, symbolically,

 \forall \varepsilon > 0 \ \ \exists \delta > 0 \ \ \forall x (0 < |x - c| < \delta \ \Rightarrow \ |f(x) - L| < \varepsilon).

A function ƒ is said to be continuous at c if it is both defined at c and its value at c equals the limit of f as x approaches c:

\lim_{x\to c} f(x) = f(c).

If the condition 0 < |x − c| is left out of the definition of limit, then requiring ƒ(x) to have a limit at c would be the same as requiring ƒ(x) to be continuous at c.

The real inequalities exploited in the above definitions were pioneered by Cauchy and formalized by Weierstrass.

It is noteworthy to add that jokes involving negative epsilon values derive their entertainment from the fact that epsilon can never be less than zero as that would imply a past x value of epsilon.

Uniform continuity

A function ƒ is said to be uniformly continuous on an interval I if

for each real ε > 0 there exists a real δ > 0 such that for all real numbers x and y in I with  |x − y| < δ, we have |ƒ(x) − ƒ(y)| <  ε,

or, symbolically,

\forall \varepsilon>0 \;\exists \delta>0 \quad \forall x\in I \;\forall y\in I \quad (|y-x|<\delta \Rightarrow |f(y)-f(x)|<\varepsilon).

While continuity is a local notion, uniform continuity is a global one.

The difference between uniform continuity on an interval, and continuity at all points in the interval separately, is that with uniform continuity, the δ that is small enough may be taken to be the same at all points in the interval. As an example sin(1/x) is a continuous function of x at every point in the interval (0, ∞), but it is not uniformly continuous on that interval.

In non-standard calculus, one defines the limit of a function and related concepts in a way that reduces the quantifier complexity of the (ε,δ)-definition and uniform continuity.

Comparison of continuity and uniform continuity in terms of quantifier formulas

Continuity of a function for every point x of an interval can thus be expressed by a formula starting with the following quantification:

\forall \varepsilon\; \forall x \;\exists \delta \;\forall y \quad (|y-x|<\delta \Rightarrow |f(y)-f(x)|<\varepsilon),

whereas for uniform continuity, the order of the second and third quantifiers is reversed:

\forall \varepsilon \;\exists \delta \; \forall x \;\forall y \quad (|y-x|<\delta \Rightarrow |f(y)-f(x)|<\varepsilon)

(the domains of the variables have been deliberately left out so as to emphasize quantifier order).

Thus for continuity at each point, one fixes a point x, and then there must exist a distance δ\forall x \;\exists \delta – while for uniform continuity a single δ must work ("uniformly") for all points x (and y): \exists \delta \; \forall x \;\forall y.

Limit of sequence

For a sequence of reals \{x_n|n\in \mathbb{N}\}\;, if L\in \mathbb{R}\; we say L is the limit of the sequence and write

 L = \lim_{n \to \infty} x_n
\Longleftrightarrow \forall \varepsilon>0\;, \exists N \in \mathbb{N}\;, \forall n \in \mathbb{N}: n>N \implies |x_n-L|<\varepsilon.\;

i.e.: if and only if for every real number ε > 0, there exists a natural number N such that for every n > N, we have |xn − L| <  ε. An alternative definition of reduced quantifier complexity appears at non-standard calculus.

Generalizations and applications

In terms of metric topology, the (ε, δ)-definition of limit can be understood as expressing convergence in terms of a basis for a topology.

The (ε, δ)-definition of limit is used in Weierstrass's definition of continuity.

See also

Notes


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