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Continuous functions 
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#1
Feb1712, 09:03 AM

P: 16

A function defined on ℝ is continuous at x if given ε, there is a δ such that f(x)f(y)<ε whenever xy<δ. Does this imply that f(x+δ)f(x)=ε? The definition only deals with open intervals so i am not sure about this. If this is not true could someone please show me a counter example for it?
Any help would be appreciated. Thanks. 


#2
Feb1712, 09:44 AM

P: 199

No. You have to learn to think differently. Draw a lot of pictures and think about limiting processes, not equalities. A real function is continuous at x if I can draw a rectangular box around the point (x, f(x)), shrink the box arbitrarily small, and the function remains in the box. Consider f(x)=1 for all x. It is continuous everywhere, but for any epsilon>0 there is no delta which satisfies your statement. In fact, f(y)f(x)=0 for all x,y, yet for any epsilon, no matter how small, I can choose delta arbitrarily large.



#3
Feb1712, 05:02 PM

Sci Advisor
P: 1,169

The description can also be interpreted as saying that one can find, for any ε>0, a value of δ>0 every point x in the interval: (yδ,y+δ) on the xaxis Is mapped into the interval (f(y)ε,f(y)+ε ) on the yaxis. Try playing with relativelysimple functions like x^{2}, and see what happens with the expression f(x+δ)f(x), for different values of δ, and how you can choose δ to make the difference be within ε. 


#4
Feb1712, 07:42 PM

P: 16

Continuous functions
Thanks for your replies. I understand now.



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