Real analysis definition question

Click For Summary

Discussion Overview

The discussion revolves around the definition of continuity in the context of real analysis, specifically focusing on the role of the function involving the metric \(\rho\) and its implications for real-valued functions. The scope includes theoretical aspects of continuity and the application of metrics in defining distances in function ranges.

Discussion Character

  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant questions the meaning of the function involving \(\rho\) in the definition of continuity.
  • Another participant explains that \(\rho\) refers to the metric (distance function) defined for the range of the function \(f\), suggesting that the definition is not limited to real functions of one variable.
  • A participant seeks clarification on whether \(\rho(f(x),b)\) represents the distance between \(f(x)\) and \(b\) in the range of \(f\), which is confirmed by another participant.
  • It is noted that while \(\rho\) can represent the absolute value for real-valued functions typically studied in calculus, other distance functions may be applicable in different contexts.

Areas of Agreement / Disagreement

Participants generally agree on the interpretation of \(\rho\) as a metric in the definition of continuity, but there is no consensus on the broader implications or specific applications of this definition beyond real-valued functions.

Contextual Notes

The discussion does not resolve the potential variations in the definition of continuity across different mathematical contexts or the implications of using different distance functions.

royzizzle
Messages
48
Reaction score
0
what does the function involving rho mean in this definition?
(in picture)
 

Attachments

  • 7 Limit of a function definition.jpg
    7 Limit of a function definition.jpg
    23.7 KB · Views: 458
Physics news on Phys.org
While the pictures demonstrate the notions of continuity and discontinuity at a real value, with graph in the plane, there is nothing in this definition that is specific to real functions of one variable. The [tex]\rho[/tex] in the definition refers to the metric (distance function) defined for the range of the function [tex]f[/tex].
 
statdad said:
While the pictures demonstrate the notions of continuity and discontinuity at a real value, with graph in the plane, there is nothing in this definition that is specific to real functions of one variable. The [tex]\rho[/tex] in the definition refers to the metric (distance function) defined for the range of the function [tex]f[/tex].

so rho(f(x),b) in this definition is the distance between f(x) and b in the range of f right?

k

thanks!
 
royzizzle said:
so rho(f(x),b) in this definition is the distance between f(x) and b in the range of f right?

k

thanks!

Yes. For real-valued functions as usually studied in calculus, both [tex]d[/tex] and [tex]\rho[/tex] are the absolute value, but in other situations other distance functions can be used.
 

Similar threads

Undergrad Questions on ##\mathbb{R}##
  • · Replies 5 ·
Replies
5
Views
2K
High School Is complex analysis really much easier than real analysis?
  • · Replies 4 ·
Replies
4
Views
2K
High School How can hyperreal numbers make infinitesimals logically sound in calculus?
  • · Replies 24 ·
Replies
24
Views
7K
Undergrad Proving Continuous Functions in Smooth Infinitesimal Analysis
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Usage of inverse function theorem in Folland
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad A correct definition of sequential right continuity of a function
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Several Questions About Smooth Infinitesimal Analysis
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Should the function f to be continuous for applying MVTI or not?
  • · Replies 4 ·
Replies
4
Views
2K
Verify property of inner product
  • · Replies 21 ·
Replies
21
Views
1K
Graduate Question about branch of logarithm
  • · Replies 2 ·
Replies
2
Views
3K