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Thread: Solving for Zero Content

  1. #1
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    Solving for Zero Content

    Note: Theorem and proposition were taken from Folland's Advanced Calculus
    Problem: Let $\displaystyle g: R \mapsto R $ be the function g(x) = x - [x], where [x] denotes the greatest integer less than or equal to x. Define
    f(x, y) = { $\displaystyle g(1/(x^2 + y^2)$ if (x,y) $\displaystyle \neq$ (0,0) ;
    0 if (x,y) = (0,0)}
    (a) Let D denote the set of points $\displaystyle (x,y) \in R^2$ at which the function f is discontinuous. Find D.
    (b) Show that a circle in the plane has zero content. (Hint: Use Proposition 4.19(c), which states If f: $\displaystyle (a_0, b_0) \rightarrow R^2$ is of Class $\displaystyle C^1$, then f([a,b]) has zero content whenever $\displaystyle a_0 < a < b < b_0$
    (c) Show that the set D has zero content.
    Remark: It follows from part (c) that f is integrable over any measurable region in the plane. See Theorem 4.21, which states the following: Let S be a measurable subset of R^2. Suppose $\displaystyle f: R^2 \mapsto R^2$ is bounded and the set of points in S at which f is discontinuous has zero content. Then f is integrable on S.

    How would I go about finding the set D for (a)? I know f is a discontinuous function but how would I find the exact (x,y)?
    Last edited by MissMousey; Nov 21st 2011 at 12:23 PM.
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  2. #2
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    Re: Solving for Zero Content

    Yeah, I'm totally lost on this problem. Hint on (a)?
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  3. #3
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    Re: Solving for Zero Content

    Quote Originally Posted by MissMousey View Post
    Note: Theorem and proposition were taken from Folland's Advanced Calculus
    Problem: Let $\displaystyle g: R \mapsto R $ be the function g(x) = x - [x], where [x] denotes the greatest integer less than or equal to x. Define
    f(x, y) = { $\displaystyle g(1/(x^2 + y^2)$ if (x,y) $\displaystyle \neq$ (0,0) ;
    0 if (x,y) = (0,0)}
    (a) Let D denote the set of points $\displaystyle (x,y) \in R^2$ at which the function f is discontinuous. Find D.
    (b) Show that a circle in the plane has zero content. (Hint: Use Proposition 4.19(c), which states If f: $\displaystyle (a_0, b_0) \rightarrow R^2$ is of Class $\displaystyle C^1$, then f([a,b]) has zero content whenever $\displaystyle a_0 < a < b < b_0$
    (c) Show that the set D has zero content.
    Remark: It follows from part (c) that f is integrable over any measurable region in the plane. See Theorem 4.21, which states the following: Let S be a measurable subset of R^2. Suppose $\displaystyle f: R^2 \mapsto R^2$ is bounded and the set of points in S at which f is discontinuous has zero content. Then f is integrable on S.

    How would I go about finding the set D for (a)? I know f is a discontinuous function but how would I find the exact (x,y)?
    So for (a), by taking into account that the result of g(x) will be some integer in Z, it is implied that $\displaystyle g(1/(x^2 + y^2))$ will be some integer in Z, so $\displaystyle 1/(x^2 + y^2))$ = q, where q $\displaystyle \in$ Z. So we write,
    $\displaystyle 1/(x^2 + y^2))$ = q $\displaystyle \rightarrow$ $\displaystyle x^2 + y^2 = 1/q$, so g($\displaystyle 1/(1/q)$) = g(q), which results into some integer in Z

    So the set of D = {(x,y) $\displaystyle \in$ R : $\displaystyle x^2 + y^2 = 1/q $}

    Is this reasonable?
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