Suppose and there is a such that . Show that .

I understand the definition of limit but I'm having trouble getting this proof started. Any help would be appreciated.

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- Oct 18th 2010, 06:23 AMzebra2147Real Analysis Limits of Functions
Suppose and there is a such that . Show that .

I understand the definition of limit but I'm having trouble getting this proof started. Any help would be appreciated. - Oct 18th 2010, 07:15 AMtonio
- Oct 18th 2010, 07:36 AMzebra2147
well... is a limit point of . And is a point that the ball is centered around but is not contained in .

- Oct 18th 2010, 08:05 PMtonio

Oh, so a punctured ball...but then the claim is false: is such that

, and nevertheless ...

Note that , and I don't care if you want to puncture it and take the middle point out or not.

It's not hard to come up with a counterexample where f is continuous, so I think something must be missing in your question...or I misunderstood, of course.

Tonio - Oct 19th 2010, 05:32 AMzebra2147
Well, I looked over his exercise and I typed it in right, and typically he doesn't try and trick us. So are you saying that your example of is continuous? Isn't that a jump discontinuity? Or am I confused?

- Oct 19th 2010, 08:00 AMtonio
- Oct 19th 2010, 09:47 AMzebra2147
Oh ok. I apologize for me misunderstanding. As far as I can see there is no information missing so maybe he left something out in his notes.

- Oct 19th 2010, 10:29 AMPlato
I rather suspect that in the OP the should be . Or visa versa.

- Oct 19th 2010, 11:58 AMtonio