
#37
May512, 04:34 PM

P: 168

In particular, your for alls need to be in terms of sequences, not elements of the sequence. 



#38
May512, 04:37 PM

P: 799

If you can "see" without any need for explanation or symbols that 1/x and e^{x} are continuous but not uniformly continuous, that's what I'm trying to explain. Because as x gets larger, e^{x} gets a LOT LOT LOT larger. And as x gets close to zero, 1/x gets a LOT LOT LOT larger. Both 1/x and e^{x} get stretched. Once you see that you can intuitively see that they must fail to be uniformly continuous. Then the epsilons and deltas will be easer. The things people were saying about differentiability, boundedness, and the compactness of the domain, are conditions that ensure that a given function is uniformly continuous. But they are not the definition or the meaning of uniform continuity. The meaning is in the stretchiness. 



#39
May512, 04:44 PM

P: 101

Well, yea, I've got the intuition now I think. e^{x} could be written as e^{x}=sinh(x)+cosh(x), because e^{x} is not uniformly continuous, then by what I proved earlier, sinh(x) 'OR' cosh(x) should be not uniformly continuous as well. Using the intuitive method, one could see that both sinh(x) and cosh(x) are in fact not uniformly continuous and it sounds reasonable because both have e^{x} in their equations that make them get steeper and steeper as we go to infinity (and also because e^{x} fails to cancel out the fast growth of e^{x} in sinh(x)). Am I right? If yes, then I've got the intuition right. 



#40
May512, 04:55 PM

P: 168

Also a bounded function can fail to be uniformly continuous. For example sin(1/x) on (0, 1). 



#41
May512, 05:02 PM

P: 101

if [itex] x_n  y_n \to 0 [/itex] then [itex] \exists M\in\mathbb{N}>0,\forall\epsilon>0,\forall n\in\mathbb{N}: n\geq M \implies f(x_n)f(y_n) < \epsilon[/itex] maybe this is where that i could come to play, because this time when I negate this there must exist n=i that has the property you said. Am I right? 



#42
May512, 05:05 PM

P: 168





#43
May512, 05:11 PM

P: 101

Because after this step I think other steps are justified, or not? 


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