Prove/Disprove: Uniform Continuity of sin(sin(x))

Click For Summary

Discussion Overview

The discussion revolves around the uniform continuity of the function \( \sin(\sin(x)) \) and the application of the \( \epsilon, \delta \) definition in proving or disproving this property. The scope includes theoretical exploration and mathematical reasoning.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant requests a proof or disproof of the uniform continuity of \( \sin(\sin(x)) \) using the \( \epsilon, \delta \) definition.
  • Several participants express confusion about the \( \epsilon, \delta \) definition, indicating a need for clarification on its meaning and application.
  • A hint is provided involving the formula \( \sin x - \sin y = 2 \cos\frac{x+y}{2} \sin\frac{x-y}{2} \) as a potential approach to the problem.
  • Another participant reiterates the \( \epsilon, \delta \) definition, attempting to clarify its components and implications for real functions.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the uniform continuity of \( \sin(\sin(x)) \). There is confusion regarding the \( \epsilon, \delta \) definition, and multiple viewpoints on how to approach the problem are present.

Contextual Notes

Some participants may lack familiarity with the \( \epsilon, \delta \) definition, which could limit their ability to engage with the problem effectively. The discussion does not resolve the mathematical steps necessary to prove or disprove uniform continuity.

solakis1
Messages
407
Reaction score
0
prove or disprove if the following function is uniformly continuous:

$$sin(sin(x)) $$ using the ε,δ definition
 
Physics news on Phys.org
Do you know what that means? What IS the "$\epsilon, \delta$ definition"?
 
Country Boy said:
Do you know what that means? What IS the "$\epsilon, \delta$ definition"?
This is a challenge problem. solaklis wants you to find the answer...

-Dan
 
hint
[sp]Use the formula :
sinx-siny =$2cos\frac{x+y}{2}sin\frac{x-y}{2}$[/sp]
 
Country Boy said:
Do you know what that means? What IS the "$\epsilon, \delta$ definition"?
The $\epsilon$ $\delta$ definition for real function f(x) is :

Given $\epsilon>0$ there exists a $\delta>0$ such that :
For all x,y belonging the to domain of f(x) and $|x-y|<\delta$ then $|f(x)-f(y)|<\epsilon$
 

Similar threads

High School Does uniform continuity of |f| imply uniform continuity of f?
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad Uniform closure of algebra of bounded functions is uniformly closed
  • · Replies 2 ·
Replies
2
Views
2K
Uniform Continuity of functions
  • · Replies 40 ·
2
Replies
40
Views
5K
MHB Does This Sequence Converge Uniformly?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Can we have a pasting lemma for uniform continuous functions
  • · Replies 2 ·
Replies
2
Views
2K
MHB Prove that series converges uniformly
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Uniform convergence of family of functions with continuous index
  • · Replies 3 ·
Replies
3
Views
2K
High School Continuity of ln(x) function
  • · Replies 11 ·
Replies
11
Views
2K
MHB Uniform Continuity and Cauchy Sequences
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Continuity of linear map from subspace of Euclidean space
  • · Replies 2 ·
Replies
2
Views
2K