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Thread: Real Analysis Limits of Functions

  1. #1
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    Real Analysis Limits of Functions

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

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

    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 $\displaystyle x_0\,\,and\,\,a$ ? What is $\displaystyle B'_r(a)$ ? ....

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

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

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

    Note that $\displaystyle (1,2)=B_{1/2}(3/2)$ , and I don't care if you want to puncture it and take the middle point $\displaystyle 3/2$ 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 $\displaystyle f(x)$ is continuous? Isn't that a jump discontinuity? Or am I confused?
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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 $\displaystyle f(x)$ 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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    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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    I rather suspect that in the OP the $\displaystyle a$ should be $\displaystyle x_0$. Or visa versa.
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    Quote Originally Posted by Plato View Post
    I rather suspect that in the OP the $\displaystyle a$ should be $\displaystyle x_0$. Or visa versa.

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

    Tonio
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