? asked in 科學數學 · 1 decade ago

微積分證明題

以下是題目圖片

http://farm3.static.flickr.com/2527/3963049571_d40...

thanks in advance!!!

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  • 1 decade ago
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    Q5.

    Define F:X -> -2X/|X| , then F is the required mapping which is a continuous function on R^n\{0} .

    For n=3, the component functions are

    F1(X)= -2x/sqrt{x^2 +y^2 +z^2}

    F2(X)= -2y/sqrt{x^2 +y^2 +z^2}

    F3(X)= -2z/sqrt{x^2 +y^2 +z^2}

    Now,

    lim_{t to 0} F( (0,0,t)) = (0, 0, -2)

    lim_{t to 0} F( (t,0,0)) = (-2, 0, 0)

    The two directional limit are not equal and hence it is impossible to define a continuous image at (0,0,0)

    (b).

    Suppose f is contiuous.

    Define g: t -> (t, 0, 0, ..., 0) , then the composite function

    G = f。g

    is a continuous function from R to R.

    lim_{t to 2+} G(t) = < 2

    lim_{t to 2-} G(t) > = 2

    Hence, G(2) =2

    However, by the defintion of f,

    G(2) = f( g(2)) = f( (2,0, ... , 0) ) > 2 since (2,0, ... , 0) belongs to S.

    The contradiction is due to the assumption "f is continuous".

    ***

    In fact, from the point of view of point set topology, the set U={x: f(x)>2} is open and hence the inverse image f^(-1)(U)=S is also open if f is continuous. But, obviously, S is not open!!

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