My Math Forum Why epsilon and delta can be changed in the definition of continuity at x
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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
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 Now we have to prove that this definition is satisfied if and only if the function is bounded by [a,b].
You don't mean bounded by [a,b], you mean bounded on the closed interval [a,b].

 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 $\delta \gt b-a$ and any $x_0$). What is the result?

 Tags changed, continuity, definition, delta, epsilon

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