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"Introduction to Analysis" by Rosenlicht


I would be very grateful if somebody could verify my proof.

Assume that $f$ is continuous at $p$. We can always choose $\delta_1>0, \delta_2>0$ such that $d'(f(x),f(p))\le\frac{1}{2n}$ whenever $d(x,p)\le\delta_1$ and $d'(f(y),f(p))\le\frac{1}{2n}$ whenever $d(y,p)\le\delta_2$. Set $\delta=\min\{\delta_1,\delta_2\}$. Then we have $d'(f(x),f(y))\le d'(f(x),f(p))+d'(f(p),f(y))\le\frac{1}{2n}+\frac{1}{2n}=\frac{1}{n}$. Thus oscillation of $f$ at $p$ is equal to $0$.

Now assume that oscillation is equal to $0$. From definition of g.l.b. for any $\epsilon>0$ there exists some open ball $B(p,\delta)$ such that for any $x,y$ in this ball we have $d'(f(x),f(y))\le \epsilon$. Choose $0<\delta'<\delta$ and fix $y=p$ and we have that for any $\epsilon>0$ there is $\delta'$ such that for any $x\in E$ it is true that $d'(f(x),f(p))\le \epsilon$ whenever $d(x,p)\le\delta'$.

The last thing to show is that for any $\epsilon>0$ the set $\{x\in E : \operatorname{osc}(f,x)\ge \epsilon \}$ is closed. It's enough to show that $S=\{x\in E : \operatorname{osc}(f,x)< \epsilon \}$ is open.

I'm not sure if the proof is correct and also I cannot figure out how to finish the last piece. Any hint please?

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Your proof seems correct to me. There are, however, some places where you could make the proof simpler. For example, in the first part you can just take $\delta_1=\delta_2=\delta$ as your two deltas are actually the same number. In the second part you don't need to introduce $\delta'$ as your statement about $\delta$ already achieves the result you want.

As for the last part, you've made a sensible start. Now it's just a case of writing out the details. Suppose $x$ is such that $\operatorname{osc}(f,x)=k< \epsilon$. If $a$ is between $k$ and $\epsilon$ then there is a $\delta$ such that for all $z_1,z_2\in B(x,\delta)$ we have $d'(f(z_1),f(z_2))<a$. Given any $x'\in B(x,\delta)$ we can find $\delta'$ such that $B(x',\delta')\subset B(x,\delta)$. Using this we can show that $\operatorname{osc}(f,x')< \epsilon$ so that $S$ is open.

I'd be interested to know where this question comes from.

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    $\begingroup$ This problem comes from "Introduction to Analysis" by Maxwell Rosenlicht. It is extremely concise book with many interesting problems. Moreover it's very cheap (Dover Books). It's the best analysis book I have come across in a while. $\endgroup$ – luka5z Dec 19 '15 at 17:19

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