What is the Epsilon-Delta Proof for Continuity?

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SUMMARY

The Epsilon-Delta proof is a formal definition of continuity for functions in calculus. Specifically, a function f is continuous at a point Xo if, for every ε (epsilon) greater than 0, there exists a δ (delta) such that if |X - Xo| < δ, then |f(X) - f(Xo)| < ε. This definition is essential for understanding the behavior of functions in mathematical analysis and is applicable to various functions within their domains.

PREREQUISITES
  • Understanding of limits in calculus
  • Familiarity with the concept of functions and their domains
  • Basic knowledge of mathematical notation and symbols
  • Experience with proofs in mathematics
NEXT STEPS
  • Study the formal definition of continuity in calculus
  • Learn how to construct Epsilon-Delta proofs for various functions
  • Explore examples of continuous and discontinuous functions
  • Investigate the implications of continuity in real analysis
USEFUL FOR

Students of calculus, mathematics educators, and anyone seeking to deepen their understanding of continuity and rigorous proof techniques in mathematical analysis.

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Homework Statement



i have a couple of questions to anser and they start 'Give epsilon - delta proofs that the following functions are continuous at the indicated points.'

im guessing its not going to be too hard but what is the name of this epsilon - delta proof so i can search for and learn about it.


The Attempt at a Solution

 
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It is just one of the definitions of Continuity.

A fuction f is continuous at Xo if

For each \epsilon &gt; 0 there exists \delta such that

|X-Xo|<\delta Implies |f(X)-f(Xo)|<\epsilon

For all X,Xo belonging to Domain of f
 

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