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Open set in $\mathbb{R}$ is a union of at most countable collection of disjoint segments

This is the theorem i need to prove;

"Let $E(\subset \mathbb{R})$ be closed subset and $f:E\rightarrow \mathbb{R}$ be a contiuous function. Then there exists a continuous function $g:\mathbb{R} \rightarrow \mathbb{R}$ such that $g(x)=f(x), \forall x\in E$."

I have tried hours to prove this, but couldn't. I found some solutions, but ridiculously all are wrong. Every solution states that "If $x\in E$ and $x$ is not an interior point of $E$, then $x$ is an endpoint of a segment of at most countable collection of disjoint segments.". However, this is indeed false! (Check Arthur's argument in the link above)

Wrong solution Q4.5;

http://www.math.ust.hk/~majhu/Math203/Rudin/Homework15.pdf

Just like the argument in this solution, i can see that $g$ is continuous on $E^c$ and $Int(E)$. But how do i show that $g$ is continuous on $E$?

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I don't know how to formulate this rigorously, but it seems, all you need to do is take the points $f(\partial E)$ and show that they can be connected by a continuous function in $\mathbb{R} \setminus Int(E)$, isnt't it? –  Karolis Juodelė Oct 23 '12 at 17:07

2 Answers 2

up vote 4 down vote accepted

A constructive and explicit proof proceeds as follows. Since $E$ is closed, $U=\mathbb{R}\setminus E$ is a countable union of disjoint open intervals, say, $U=\bigcup (a_n,b_n)$. Necessarily, we must have that $a_n,b_n\in E$. Define $f(x)$ as follows. $$ f(x) = \begin{cases} g(x) &\text{if }x\in E \\ \frac{x-a_n}{b_n-a_n}g(b_n)+\frac{b_n-x}{b_n-a_n}g(a_n) & \text{if }x\in[a_n,b_n] \end{cases} $$

Notice first that $f(x)$ is well-defined and also, for all $x\in(a_n,b_n)$, either $g(a_n)\le f(x)\le g(b_n)$ or $g(b_n)\le f(x)\le g(a_n)$ depending on whether $g(a_n)\le g(b_n)$ or otherwise. Clearly, $f$ is continuous on $U$. Now suppose that $x\in E$ and $\epsilon>0$. Then there are a few cases.

Case 1: Suppose that for every $\eta>0$, $(x-\eta,x)\cap E\not=\emptyset$ and $(x,x+\eta))\cap E\not=\emptyset$. Then since $f\vert_E=g$, there is some $\delta>0$ such that if $y\in E$ and $\vert x-y\vert<\delta$ then $\vert f(x)-f(y)\vert<\epsilon$. Because of the condition we have for Case 1, we may choose some $x_1,x_2\in E$ with $x-\delta<x_1<x<x_2<x+\delta$. Choose $\delta'=\min\{x-x_1,x_2-x\}$. If $\vert y-x\vert<\delta'$, then if $y\in E$, we're done. If $y\in U$, then $y\in(a_m,b_m)$ for some $m\in\mathbb{N}$. Furthermore, $a_m,b_m\in E$ and are within $\delta$ of $x$. Also, $f(y)$ is lies between $g(a_m)$ and $g(b_m)$. Thus $f(y)$ is within $\epsilon$ of $f(x)$ since $f(a_m)=g(a_m)$ and $f(b_m)=g(b_m)$ are within $\epsilon$ of $f(x)$.

Case 2: There is some $\eta>0$ for which $(x-\eta,x)\cap E=\emptyset$ or $(x,x+\eta)\cap E=\emptyset$. In this case, $x$ is an endpoint of one of the intervals of $U$. Thus $f$ is linear on either $[x,x+\eta)$ or $(x-\eta,x]$ (maybe both). Certainly, we can get a $\delta>0$ corresponding to $\epsilon$ on this side of $x$. For the other side of $x$, use the argument from Case 1 to get some $\delta'$. Choosing $\delta''=\min\{\delta,\delta'\}$ proves the result.

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I'm afraid your proof is wrong. In case 2, $x$ need not be an endpoint. You can check this in the link in my post. –  Katlus Oct 23 '12 at 18:07
    
@Katlus, my apologies, I switched $\not=$ and $=$. It's fixed now. –  J. Loreaux Oct 23 '12 at 18:37
    
@Katlus: The argument is incomplete, though the hole is easily patched. As it stands, it works only if $E$ is unbounded in both directions. If $E$ is bounded above, let $f(x)=f(\sup E)$ for $x\ge\sup E$, and similarly if $E$ is bounded below. –  Brian M. Scott Oct 23 '12 at 18:47
    
@Brian, that's a valid point. Thanks for mentioning it. –  J. Loreaux Oct 23 '12 at 18:51
    
You’re welcome. –  Brian M. Scott Oct 23 '12 at 18:51

This is a special case of the Tietze extension theorem. This is a standard result whose proof can be found in any decent topology text. A rather different proof can be found here.

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3000 answers, Prof.! –  Pedro Tamaroff Oct 23 '12 at 17:41
    
@Peter: So it is; I hadn’t any idea that I’d written so many. –  Brian M. Scott Oct 23 '12 at 17:54
    
@Brian Yes, Tietze extension theorem is a generalization of this theorem. I searched for it, but it seems like it's almost impossible to prove that theorem on my level right now, so I wanted to prove it at least for $\mathbb{R}$ which is a special case just for now. –  Katlus Oct 23 '12 at 18:01

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