# 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?

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 $$d_Y(f(x),q) for all points $$forall x in D$$ for which $$0. 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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