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Math Help - Real Analysis Limits of Functions

  1. #1
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    Real Analysis Limits of Functions

    Suppose lim_{x\longrightarrow x_{0}}f(x)=b and there is a r>0 such that f(B'_{r}(a)) \subseteq [0,\infty). Show that b\geq 0.

    I understand the definition of limit but I'm having trouble getting this proof started. Any help would be appreciated.
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  2. #2
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    Quote Originally Posted by zebra2147 View Post
    Suppose lim_{x\longrightarrow x_{0}}f(x)=b and there is a r>0 such that f(B'_{r}(a)) \subseteq [0,\infty). Show that b\geq 0.

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


    What is the relation between x_0\,\,and\,\,a ? What is B'_r(a) ? ....

    Tonio
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  3. #3
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    well... x_{0} is a limit point of f. And a is a point that the ball B'_{r}(a) is centered around but a is not contained in B'_{r}(a) .
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  4. #4
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    Quote Originally Posted by zebra2147 View Post
    well... x_{0} is a limit point of f. And a is a point that the ball B'_{r}(a) is centered around but a is not contained in B'_{r}(a) .

    Oh, so a punctured ball...but then the claim is false: f(x)=\left\{\begin{array}{rl}-1&,\,if\,\,\,x\leq 0\\2&,\,if\,\,\,x>0\end{array}\right. is such that

     f(x)\xrightarrow [x\to -1]{}-1\,,\,\,and\,\,\,f((1,2))\subset [0,\infty) , and nevertheless b=-1<0 ...

    Note that (1,2)=B_{1/2}(3/2) , and I don't care if you want to puncture it and take the middle point 3/2 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
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    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 f(x) is continuous? Isn't that a jump discontinuity? Or am I confused?
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  6. #6
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    Quote Originally Posted by zebra2147 View Post
    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 f(x) is continuous? Isn't that a jump discontinuity? Or am I confused?

    No, it is not continuous. What I said is that's easy to give a continuous counterexample to your claim, and that's why I thought some data must be missing.

    Tonio
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  7. #7
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    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.
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  8. #8
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    I rather suspect that in the OP the a should be x_0. Or visa versa.
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  9. #9
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    Quote Originally Posted by Plato View Post
    I rather suspect that in the OP the a should be x_0. Or visa versa.

    That's exactly what I thought, but when I asked I was told all is fine...if a=x_0 then the claim is straightforward.

    Tonio
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