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 January 31st, 2012, 01:14 AM #1 Member   Joined: Jan 2012 Posts: 63 Thanks: 0 continuous and intermediate value theorem assume f:R->R has this property A at c when, for each ?>0 there exists a ??0, s.t. |x-c|?? implies |f(x)-f(c)|??. Show that every function has this property at every c, c is real number (i know when ?=0 ,then x=c we have f(x)-f(c)=0
 February 7th, 2012, 10:16 PM #2 Newbie   Joined: Feb 2012 Posts: 1 Thanks: 0 Re: continuous and intermediate value theorem I think the point of this problem is that you almost have the definition of continuity at the point 'c'. The change is "??0", which allows you to choose ?=0 and the condition |x-c|??=0 just becomes x=c. Then f(x)=f(c) and you're done. The point of this exercise is just to notice why we need ?>0 in the definition of continuity.

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