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Math Help - continuous functions

  1. #1
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    continuous functions

    These two problems appear in my textbook:

    Give a function f: R to R continuous only at x = 0 and x = 1.

    For this, let f(x) = x^2 if x is rational and x if x is irrational.
    Then it is continuous only at x^2 = x which implies x equals 0,1.

    Give a function f: R to R continous only at x1, x2,....,xn.
    This one I'm not really sure what to do. Can you do something similar to the first one?
    Any help would be appreciated.
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  2. #2
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    Consider the polynomial,
    f(x)=(x-1)...(x - n) is you multiply it out you get some negative terms and some positive terms. So we can write f(x) = p(x) - n(x) (and now n(x) has only positive terms).

    Example: Say f(x) = (x-1)(x-2) = x^2 - 3x + 2 then p(x) = x^2 + 2 and n(x) = 3x.

    Now define g(x) to be p(x) at rational points and n(x) at irrational points.

    Example: As above define g(x) to be x^2 + 2 at the rational points and 3x at the irrational points.

    Then it is continous precisely when p(x) = n(x) \implies p(x) - n(x) = 0 so f(x) = 0 so (x-1)...(x-n) = 0 so 1,2,...,n. This mean it is continous at exactly these n points.

    Note: It is not really necessary to do what I did there are many ways to construct such functions but I did it because it looks more pleasing.
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  3. #3
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    I'm not 100% sure what you're doing.

    What would the function look like?

    Could it be:
    f(x) = (x - x1)(x - x2)....(x - xn) if x is rational and 0 if x is irrational?
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  4. #4
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    Quote Originally Posted by MKLyon View Post
    I'm not 100% sure what you're doing.
    Here is a slight variation of TPH’s function (it is the same idea really).
    f(x) = \left\{ {\begin{array}{rr}<br />
   {\left| {\left( {x - x_1 } \right)\left( {x - x_2 } \right) \cdots \left( {x - x_n } \right)} \right|} & {x \in Q}  \\<br />
   { - \left| {\left( {x - x_1 } \right)\left( {x - x_2 } \right) \cdots \left( {x - x_n } \right)} \right|} & {x \in \Re \backslash Q}  \\<br />
\end{array}} \right..

    Do you see that it is continuous on \left\{ {x_1 ,x_2 , \cdots ,x_n } \right\}.
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