Re: [分析] Composition of functions

[dorminia (重新出發)]

Only if part:

                       -1
Step 1. Consider G = {g  (x)|x is in g(D)}, which is a partition of D.

Step 2. For each element A in G, choose an element x_A tha is in A.

Step 3. Define h: g(D) -> f(D) by:

           h(y) = f(x_{g-1(y)})

        Then, by the assumption g(x_1)=g(x_2) => f(x_1)=f(x_2), h is well-

        defined and uniquely determined independent to the choice of x_A's


Thus, f(x) could be writen as f(x) = h(g(x)), which is a function of g.



※ 引述《yaochia (Well, well done)》之銘言：
: 請教一個證明：
:   Given two functions f, g on the same domain D, then
:   f is a function of g  iff  for all x_1, x_2 in D,
:                              we have g(x_1) = g(x_2) => f(x_1) = f(x_2)
: 由左到右由 composite function 的定義就可以 show 出
: 但對於如何從右到左我至今一點頭緒都沒有
: 還請板上的前輩指點 orz

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