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Math Help - continuity for a real-valued function

  1. #1
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    continuity for a real-valued function

    I have one more question for now at least. My task is as follows.

    Let  f \in \mathbb{R} \wedge \text{be continuous } \backepsilon dom f = (a,b). Show that if  f(r) = 0 \, \text{for each } r \in \mathbb{Q} \backepsilon r \in (a,b), \text{then } f(x) = 0 \forall x \in (a,b).
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    Use that the rationals are dense in \mathbb{R} by chusing for every x in the domain, a sequence of rational numbers x_n in the domain converging to x. Since f is continuous, \lim_{n\to \infty} f(x_n) = f(\lim_{n\to \infty}x_n), and since the LHS is zero, the RHS is too.
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    I have heard of the topic of `denseness', but we have not formally discussed this concept in my analysis class. Is there another way of framing your reply, or perhaps could you restate the reply while including the definition of denseness???

    Thanks,
    -the Doctor
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    Quote Originally Posted by thedoctor818 View Post
    I have heard of the topic of `denseness', but we have not formally discussed this concept in my analysis class. Is there another way of framing your reply, or perhaps could you restate the reply while including the definition of denseness???
    In order to do this particular problem your class must have proven that:
    between any two real numbers there is a rational number and an irrational number.
    Have you proved that?
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    We have talked about this sort of informally, but not very formally or in an `official' manner.
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    Quote Originally Posted by thedoctor818 View Post
    We have talked about this sort of informally, but not very formally or in an `official' manner.
    In that case, You must prove the following two therorems.
    Between any two real numbers there is a rational number.
    Between any two real numbers there is an irrational number..
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    Please help. How do I prove or begin to prove those aforementioned theorems.
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    Quote Originally Posted by thedoctor818 View Post
    Please help. How do I prove or begin to prove those aforementioned theorems.
    Try this
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    Quote Originally Posted by thedoctor818 View Post
    Please help. How do I prove or begin to prove those aforementioned theorems.
    Have you covered the Archimedean Property? It states that

    (a) For every real number x, there exists a natural number n such that n > x.
    (b) Similarly, for every real number y, there exists a natural number m such that 1/m < y.

    (a) Should be trivial. (b) Follows from (a) (raise both sides of the inequality to the (-1), and of course reverse the inequality).
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    So, if I have this correctly (in order to prove there exists a rational number between x & y):

    Let  x,y \in \mathbb{R} \backepsilon x < y and consider  (x,y). We must find a rational number inside this interval. By the Archimedean Theorem,  \exists n \in \mathbb{N} \backepsilon n > \dfrac{1}{y-x}. \Rightarrow  ny > nx + 1. We then find an  m \in \mathbb{Z} \backepsilon m \leq nx + 1 < m + 1. Now, we have that  m-1 \leq nx < ny. \Rightarrow  x < \dfrac{m}{n} \leq x + \dfrac{1}{n} < y, thus proving an  \dfrac{m}{n} \in  \mathbb{Q} \backepsilon \dfrac{m}{n} \in (x,y).


    Is that correct? If so, I am uncertain of how to prove this for  a \in \overline{\mathbb{Q}}.
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    Quote Originally Posted by thedoctor818 View Post
    So, if I have this correctly (in order to prove there exists a rational number between x & y):

    Let  x,y \in \mathbb{R} \backepsilon x < y and consider  (x,y). We must find a rational number inside this interval. By the Archimedean Theorem,  \exists n \in \mathbb{N} \backepsilon n > \dfrac{1}{y-x}. \Rightarrow  ny > nx + 1. We then find an  m \in \mathbb{Z} \backepsilon m \leq nx + 1 < m + 1. Now, we have that  m-1 \leq nx < ny. \Rightarrow  x < \dfrac{m}{n} \leq x + \dfrac{1}{n} < y, thus proving an  \dfrac{m}{n} \in  \mathbb{Q} \backepsilon \dfrac{m}{n} \in (x,y).


    Is that correct? If so, I am uncertain of how to prove this for  a \in \overline{\mathbb{Q}}.
    How about this. To prove that the Rationals (Q) are dense in the Reals (R), we must show that for two real numbers x and y, there exists a rational number q = m/n satisfying x < m/n < y.

    By (a) we know that an m exists such that m > nx, which implies that x < m/n, the left half of the inequality. By (b) we know that an n exists such that 1/n < y/m, which implies that m/n < y, the right half of the inequality. Thus, there exists a rational number q = m/n x < m/n < y for any two arbitrary real numbers x and y.

    For the irrationals, choose an irrational number w, and add w to x, m/n, y. Then, prove that m/n + w is no longer rational.
    Last edited by davismj; February 23rd 2010 at 10:07 AM. Reason: Stupid mistakes that I make on every exam.
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    Thanks, I'll try that first thing in the morning. Once I have done that, however; I am not quite certain of the relationship of these 2 proofs to my problem though???
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    Quote Originally Posted by thedoctor818 View Post
    Thanks, I'll try that first thing in the morning. Once I have done that, however; I am not quite certain of the relationship of these 2 proofs to my problem though???
    Never seen that kind of question before (I'm in my undergraduate Real Analysis class at the moment). From what I can tell, it follows from the fact that for any irrational number y, there are two rational numbers p and q satisfying p < y < q. Since f(p) = f(q) = 0 and f(x) is continous on (a,b), f(y) must be 0 as well.

    As for a formal proof, sorry.
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  14. #14
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    I am also in my undergrad analysis class, & this is one of our problems. Thanks again for your help, & for the last bit - your intuition - maybe we can rework that into a formal proof. Thanks.
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    Quote Originally Posted by davismj View Post
    Never seen that kind of question before (I'm in my undergraduate Real Analysis class at the moment). From what I can tell, it follows from the fact that for any irrational number y, there are two rational numbers p and q satisfying p < y < q. Since f(p) = f(q) = 0 and f(x) is continous on (a,b), f(y) must be 0 as well.

    As for a formal proof, sorry.
    Quote Originally Posted by thedoctor818 View Post
    I am also in my undergrad analysis class, & this is one of our problems. Thanks again for your help, & for the last bit - your intuition - maybe we can rework that into a formal proof. Thanks.
    What it means for \mathbb{Q} to be dense in \mathbb{R} is that ever point of \mathbb{R} is a limit point of \mathbb{Q}. Consequently, it easy then to extract a rational sequence \{q_n\} such that q_n\to x for ever  x\in \mathbb{R}. So, since your function is continuous we have that x_n \to x\implies f(x_n)\to f(x) and so let x\in (a,b) then there exists some q_n\to x and so f(x)=\lim\text{ }f(q_n)=\lim\text{ }0=0

    Or, maybe more understandable.

    Suppose that f(x)\ne 0 for some x\in(a,b). and let \varepsilon=\frac{|f(x)|}{2}>0. Since f is continuous we should be able to find some B_{\delta}(x) such that f\left(B_{\delta}(x)\right)\subseteq B_{\varepsilon}(f(x))\subseteq \mathbb{R}^+. But, no matter what B_{\delta}(x) you pick there will be some q\in\mathbb{Q} such that q\in B_{\delta}(x) and so f(q)=0\in f\left(B_{\delta}(x)\right)\notin (0,\infty)
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