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January 31st, 2012, 02:14 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. xc?? 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<?, but how about f(x)f(c)=? i don't know how to start it )

February 7th, 2012, 11:16 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 xc??=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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continuous, intermediate, theorem 
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