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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 Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post dtn_perera Calculus 2 August 9th, 2013 06:42 PM dtn_perera Calculus 2 August 9th, 2013 06:46 AM kevinnn Calculus 7 February 20th, 2013 08:40 PM Saviola Calculus 11 October 28th, 2010 06:31 AM ryuken3k Calculus 1 October 6th, 2008 07:24 PM

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