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I was asked as a homework problem to decide whether for an interval $[a,b]$ and measurable function $f:[a,b] \to \mathbb{C}$ there exists for every $\varepsilon>0$ a compact subset $K$ such that $\lambda([a,b]\backslash K)< \varepsilon$ and such that $f$ restricted to $K$ is continuous ($\lambda$ is the usual Lebesgue measure). I thought of attacking this problem by letting $K$ be a suitably dense fat Cantor set, the denseness depending on $\varepsilon$. My motivation is as follows:

The Cantor set (and unless I'm mistaken, all fat Cantor sets too) is totally disconnected, hence around every point in the set there is some small open neigborhood separating it from the rest of the set. Then any singleton consisting of a point in the Cantor set will be open in the subspace topology, and any function from the Cantor set will thus be forced to be continuous.

Thus, by just choosing a Cantor set fat enough, I will have proved the problem in the positive (and even skipped the measurability condition on the function $f$). The main question of this post is thus: does my reasoning above make sense? Or have I missed something?

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    $\begingroup$ Probably the intention is that you take a look at Lusin's theorem. $\endgroup$ Mar 9, 2016 at 13:12
  • $\begingroup$ Ok, thanks for the hint. $\endgroup$
    – MonadBoy
    Mar 9, 2016 at 13:13
  • $\begingroup$ Oh, upon looking up Lusin's theorem, it literally says exactly what was asked for in the problem…shucks… $\endgroup$
    – MonadBoy
    Mar 9, 2016 at 13:27

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Yes, the Cantor set is totally disconnected. It does not follow (and it's not true) that every point is isolated. Not every function defined on the Cantor set is continuous.

You could use a Cantor-set-ish construction. This being just a hint, suppose $0\le f < 1$. Let $A_0$ be the set where $0\le f<1/2$ and $A_1$ the set where $1/2\le f<1$.

Choose compact sets $K_j\subset A_j$ of almost full measure. Now let $A_{0,0}=\{x\in K_0:0\le f(x)<1/4\}$ and $A_{0,1}=\{x\in K_0:1/4\le f<1/2\}$...

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  • $\begingroup$ Ok, thanks for the hint. Could you also, for the sake of completeness, give me an example of a discontinuous function from the Cantor set, so I can see where I went wrong? $\endgroup$
    – MonadBoy
    Mar 9, 2016 at 13:17
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    $\begingroup$ @A.Sh, map $0$ to $0$ and everything else to $1$. $\endgroup$
    – Carsten S
    Mar 9, 2016 at 13:20
  • $\begingroup$ @CarstenS Ok, thanks for the help. $\endgroup$
    – MonadBoy
    Mar 9, 2016 at 13:21
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    $\begingroup$ @A.Sh What Carsten said. Look, you know that $0$ is in the Cantor set and that $1/3^n$ is also in the Cantor set for $n=1,2,\dots$. So $0$ is not an isolated point (in fact sort of the whole point to the Cantor set is it's totally disconnected with no isolated point). $\endgroup$ Mar 9, 2016 at 13:22
  • $\begingroup$ Ah, of course. Thanks for setting me straight. $\endgroup$
    – MonadBoy
    Mar 9, 2016 at 13:23

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