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November 3rd, 2015, 09:12 AM  #1 
Newbie Joined: Nov 2015 From: Italy Posts: 3 Thanks: 0  Why epsilon and delta can be changed in the definition of continuity at x
We have a function f:[a,b]→R, where [a,b] is an interval bounded by the real straight line. Now someone writes the definition of continuity at x∈[a,b]x∈[a,b] but makes a mistake. He writes δ in stead of ε and ε instead of δ. So the definition changes: for every δ>0 there exists ε>0 such that if x∈[a,b],x−x0<δ then f(x)−f(x0)<ε Now we have to prove that this definition is satisfied if and only if the function is bounded by [a,b]. How can I prove this? 
November 3rd, 2015, 09:29 AM  #2  
Senior Member Joined: Jun 2015 From: England Posts: 915 Thanks: 271 
Suggestion: Quote:
 
November 3rd, 2015, 09:37 AM  #3 
Newbie Joined: Nov 2015 From: Italy Posts: 3 Thanks: 0 
Yes sry

November 3rd, 2015, 09:54 AM  #4 
Senior Member Joined: Jun 2015 From: England Posts: 915 Thanks: 271 
OK so the usual way to proove if and only if is to start with the contrapositive (imagine it is / is not true and show that leads to a contradiction. Why do you think the boundedness matters, what would happen if were not bounded?

November 3rd, 2015, 10:33 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra 
For "bounded $\implies$ satisfied", identify a suitable $\epsilon$. For "satisfied $\implies$ bounded", follow studiot's idea. 
November 3rd, 2015, 12:42 PM  #6 
Newbie Joined: Nov 2015 From: Italy Posts: 3 Thanks: 0 
Can you please be more explicit. I really don't get it.

November 3rd, 2015, 01:45 PM  #7 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra 
If a function is bounded on $[a,b]$, then there exists an $M \gt 0$ such that $M \lt f(x) \lt M$ for all $x \in [a,b]$. If the conditions of the statement are satisfied, then we may pick $x_0 = \frac12(a + b)$ and $\delta \gt \frac12(a  b)$. (Or just pick $\delta \gt ba$ and any $x_0$). What is the result? 

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changed, continuity, definition, delta, epsilon 
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