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F: [0,infinity) > R is continuous at every point of its domain 
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#1
Nov1609, 02:22 PM

P: 1

Problem:
Assume that f: [0,infinity) > R is continuous at every point of its domain. Show that if there exists a b>0 so that f is uniformly continuous on the set [b,infinity) then f is uniformly continuous on [0,infinity). I don't really know where to start with this one, any help would be greatly appreciated 


#2
Nov1609, 03:40 PM

Sci Advisor
P: 6,040

The main point would be to prove that if f is continuous on [0,b], it will be uniformly continuous on that interval, and therefore on [0,∞).



#3
Nov1709, 02:06 AM

P: 1,635

Like said, if you haven't learned compactness,then you can prove it by contradiction. that is, start by assuming that even though f is continuous on (lets take a more general case) [a,b], it is not uniformly continuous on [a,b]. What does this mean in terms of the definition of uniform continuity?? Try to generate two sequences a_n and b_n, such that a_nb_n<d, implies f(a_n)f(b_n)>=e, and come to a contradiction somewhere along those lines. This theorem, (without using the notion of compactness, was part of one of my projects for HOnors Real Analysis, so if interested, let me know, i can email the whole project to you). But try it first on your own. Cheers! 


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