# 微積分證明題

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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".

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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!!