How prove $h(x)=f(x)-g(x)$

Question

show that: every function $h(x):R\to R$ can be written as $$h(x)=f(x)-g(x)$$ where $f(x),g(x)$ are satisfying the intermediate value property：

http://en.wikipedia.org/wiki/Intermediate_value_theorem

My try: since $f(x),g(x)$ are all such Intermediate value theorem

mean that:

so for $f(x)$,and for any $[a,b]$, there exsit $\xi\in(a,b)$,such $$f(\xi)=\eta,$$ where $f(a)<\eta<f(b)$

similar for $g(x)$,and for any $[c,d]$,there $\xi_{1}\in[c,d]$,such $$f(\xi_{1})=\eta_{1}$$ where $\eta_{1}\in (f(c),f(d))$

and Now How can for any function $h(x)$,then we always can $$h(x)=f(x)-g(x)$$ Thank you

The problem is solved on page 77 of Andreescu and Gelca, Mathematical Olympiad Challenges. — Gerry Myerson, Dec 28 '13 at 02:52
@copper.hat: The meaning must be that $f$ and $g$ have the intermediate value property (aka they are "Darboux functions"). — hmakholm left over Monica, Dec 28 '13 at 02:53

Salech Alhasov · Answer 1 · 2013-12-28T03:38:42.477

1

That is Sierpinski’s Theorem and you can find it on Beni Bogoşel blog over here.

(For your case, you need redefine $f_1$ and $f_2$ that mentioned there.)

Edit:

Theorem 3 in: Darboux functions - Beni Bogoşel.pdf

edited Dec 28 '13 at 03:38

answered Dec 28 '13 at 03:24

Salech Alhasov

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Hello,you link this blog,I can't open it,Thank you – math110 Dec 28 '13 at 03:29
@math110: It is basically the same construction as the one I hint at in my answer. – hmakholm left over Monica Dec 28 '13 at 03:32
I find this book have proof: http://books.google.com.hk/books?id=fXfEG-F2zJUC&pg=PA6&redir_esc=y&hl=zh-CN&sourceid=cndr#v=onepage&q&f=false – math110 Dec 28 '13 at 03:39
@math110: see the link to pdf file. – Salech Alhasov Dec 28 '13 at 03:39
@SalechAlhasov,Thank you,also can't open it – math110 Dec 28 '13 at 04:00

score 0 · Answer 2 · answered Dec 28 '13 at 02:59

Using the axiom of choice, you can achieve something stronger, namely that the image of any nonempty interval under $f$ and $g$ is all the real numbers.

Hint: Start by choosing a basis for $\mathbb R$ as a vector space over $\mathbb Q$. Then partition $\mathbb R$ into two dense locally continuum-sized sets $A$ and $B$, and make sure that $f$ will take all values on $(x,y)\cap A$ for any $x<y$ and $g$ will take all values on $(x,y)\cap B$ for any $x<y$. Then choose the values of $f$ on $B$ and $g$ on $A$ such that their difference is the desired $h$.

How prove $h(x)=f(x)-g(x)$

2 Answers2