Surjectiveness in a compact subset

Question

I am completely at a loss as how to proceed. I can't use differentiability here.The question is

Let $K$ be a compact subset of $\mathbb{R}$ and $f:K\rightarrow K$ be a function satisfying the condition $|f(x)-f(y)|=|x-y|\ \ \ \ \ \forall \ x,y\in K$ . Show that $f$ is surjective.

Please help..

@JohnMartin Please provide an elaborate answer. To prove $f$ to be continuous, I think a lot many cases have to be proved independently. — AbracaDabra, Apr 12 '16 at 20:01
No, all you need is to say that for any $\epsilon>0$ there exists a $\delta > 0$ such that $|x-y|<\delta \Rightarrow |f(x) - f(y)|<\epsilon$. But this special property $|f(x) - f(y)| = |x-y|$ means we can take $\delta = \epsilon$. — John Martin, Apr 12 '16 at 20:03
And now you can say something about a continuous function on a compact set... — Jürgen Sukumaran, Apr 12 '16 at 20:03
Do you have to prove, that an isometry on itself of a compact subspace of a line is surjective? — Édes István Gergely, Apr 12 '16 at 20:04
@ÉdesIstvánGergely :Can you please elaborate your question? — AbracaDabra, Apr 12 '16 at 20:11

score 1 · Answer 1 · answered Apr 12 '16 at 20:14

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Because $K$ is a compact subset of $\mathbb{R}$ it has a minimum, $a$, that every element of $K$ is not less than $a$, and a maximum too $b$. These are the only points in $k$ that their distance is the diameter of $K$, so $f$ maps the set $\{a,b\}$ into $\{a,b\}$. You can show (or you can use the fact, that there are only two kind of isometries of $\mathbb{R}$: translation and reflection on a point), if $f(a)=a$ then $f$ is the identity, and if $f(a)=b$ then $f(x)=a+b-x$.

answered Apr 12 '16 at 20:14

Édes István Gergely

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Please elaborate more. I get what you stated. But how to show that there are only 2 kinds of isometries? – AbracaDabra Apr 12 '16 at 20:17
First you show, that if you know where an isometry maps two points, it is determined. It's easy: if you know the distance of a point of the real line from two points, you know the point. On the other hand you have to show, that if you know the image of a point by an isometry of the real line, every point has two option where to get mapped by that isometry. Also it's easy, there are only two point a given distance from a given point. – Édes István Gergely Apr 12 '16 at 20:21

Surjectiveness in a compact subset

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