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Why in the ε-δ definition of a limit is x∈D instead of just ℝ

Mathematics Asked by Tiffany on December 25, 2020

ε-δ definition:

$(forall epsilon >0) (exists delta >0)(forall x in D) (0 < lvert x – a rvert < delta Rightarrow lvert f(x) – L rvert < epsilon)$

I don’t understand why in the epsilon-delta limit definition x is defined as being x∈D instead of just using x∈ $mathbb{R}$. Would specific issues arise for certain functions or types of functions if I were to use x∈ $mathbb{R}$ instead?

2 Answers

I would like suggest as answer conception of limit with respect to some set D in metrical spaces, for example Rudin W. - Principles of mathematical analysis,1976, 83-84p. :

4.1 Definition. Let $X$ and $Y$ be metric spaces; suppose $D subset X$, $f$ maps $D$ into $У$, and $p$ is a limit point of $D$. We write $f(x) to q$ as $x to p$, or $$limlimits_{x to p}f(x)=q$$ if there is a point $qin Y$ with the following property: For every $varepsilon >0$ there exists a $delta > 0$ such that $d_Y(f(x),q)<varepsilon$ for all points $forall x in D$ for which $0<d_X(x,p)<delta$. The symbols $d_X$ and $d_Y$ refer to the distances in $X$ and $Y$, respectively.

If $X$ and/or $Y$ replaced by the real line, the complex plane, or euclidean space, the distances are replaced by absolute values or by norms.

Answered by zkutch on December 25, 2020

I think that $D$ is the domain of the function. Take for example the function $f(x)=ln(x)$.

You know that this function is a map between $underbrace{(0,infty)}_Dtomathbb R$, so in this case it does not make sense to study continuity at a point $x = -2$ ,for example, as this function is not defined here. If you have a function $f: mathbb R to mathbb R$, for example $f(x)=x$, then your $D = mathbb R$ and then you can let $x in mathbb R$.

Answered by Eduardo Magalhães on December 25, 2020

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