MQ52 --- Function

2012-11-01 10:57 pm
MQ52 --- Function Difficulty: 70%Let f: ℝ → ℝ be a function that satisfyf(x + y) ≤ yf(x) + f(f(x)) for all x, y ∈ ℝ.Prove that f(x) = 0 for all x ≤ 0.

回答 (1)

2012-11-03 10:21 pm
✔ 最佳答案
http://i1090.photobucket.com/albums/i376/Nelson_Yu/int-668.pnghttp://i1090.photobucket.com/albums/i376/Nelson_Yu/int-1357.png
圖片參考:http://i1090.photobucket.com/albums/i376/Nelson_Yu/int-668.png


圖片參考:http://i1090.photobucket.com/albums/i376/Nelson_Yu/int-1357.png



2012-11-10 16:47:00 補充:
I do not understand your question.
Since b is -ve finite, b/(n+1) is a small -ve value.
And since f(x) is a non-negative function, I concluded that f(f(-n)) is bigger than a small -ve value and also non-negative, so itself muse be a small -ve value.
There is no relation to what a is!!!

2012-11-11 00:53:51 補充:
I do not need to know the case when n is not >>0
I want to prove f(-infinity) = 0, that's all

2012-11-11 00:57:02 補充:
as far as n is large enough, f(-n) is close to 0.
(**) says it is either exactly 0 or less than -n
since it is close to 0, then it is exactly 0

2012-11-11 11:47:35 補充:
I have rewritten the second part of the proof. Please see if it is easier to understand.

2012-11-12 13:36:47 補充:
look at this:
http://i1090.photobucket.com/albums/i376/Nelson_Yu/int-1960.png


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