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asked in 科學數學 · 1 decade ago

高等微積分....急

Suppose that the function f:[0,無窮大) →[0,無窮大)is continuous and

stable and f(0)=0,Prove that f:[0,無窮大) →[0,無窮大) is one-to-one

and onto.

麻煩請寫詳細一點..

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  • 1 decade ago
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    ( one to one)

    f: [0,∞) -> [0, ∞) is stable, then

    there exists a positive constant c , such that

    | f(x) - f(y) | >= c |x-y| for all x, y >=0

    ( 1-1 )

    If f is not one-to-one, then there exists two numbers x1, x2, such that f(x1)=f(x2)

    => | f(x1) - f(x2) | = 0 >= c |x1-x2| contradiction.

    (onto)

    Given any positive number r,

    (1) f(0)=0

    (2) f( (r+1)/c ) = f( (r+1)/c ) - f(0) >= c* (r+1)/c = r+1> r

    By intermediate value theorem of continuous function, there exists

    a number x in (0, (r+1)/c) such that f(x)= r.

    So, f(x) can take any positive value, i.e. f:[0, ∞)->[0, ∞) is onto.

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