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January 31st, 2012, 02:14 AM   #1
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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<?, but how about f(x)-f(c)=? i don't know how to start it )
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February 7th, 2012, 11:16 PM   #2
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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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