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I always thought, that integrability of a function is a global property while differentiability is a local property. After reading What's the definition of a "local property"? I doubt my opinion. In the linked MSE question Najib Idrissi gives the following definition of a local property:

a property $P$ is local when, for all spaces $X$, if $\{U_i\}$ is an open cover of $X$ and all the $U_i$ have $P$, then $X$ has $P$.

Now I think that the property, whether a function $f:[a,b]\to\mathbb R$ is integrable, fulfills this definition. If $\{U_i\}$ is an open cover of $[a,b]$ and $f|_{U_i}$ is integrable for all $U_i$, then also $f:[a,b]\to\mathbb R$ is integrable since $[a,b]$ is compact.

My Question: Am I right, that he property "The function $f:[a,b]\to\mathbb R$ is integrable." is a local property of $f$? If not: Where have I made a mistake?

Note: I restrict the domain of $f$ to a closed interval $[a,b]$. I am aware, that the above argumentation does not hold for functions with arbitrary domains. For example $f:(0,1)\to\mathbb R : x \mapsto \frac 1x$ is not integrable but one can find an open cover $\{U_i\}$ of $(0,1)$ where all $f|_{U_i}$ are integrable.

Update: It might be an explanation, that we have to use another definition of a local property for a function. See What is the definition of a local property for a function? with more details...

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Stephan Kulla
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    At first sight, it seems that “integrability on a compact space” might be a local property. Now compactness makes many local things global. – Pedro Sánchez Terraf Feb 15 '16 at 18:31
  • @PedroSánchezTerraf: You mean "compactness makes many global things local"?! – Stephan Kulla Feb 15 '16 at 18:34
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    This was a hand-waiving around the fact that many local properties fail to hold in the whole space $X$ because you need infinitely many open sets to cover it. Now compactness lets you always choose some finite subcover to put together all the pieces of local information. For a rather uninteresting example, consider "$f:X\to\mathbb{R}$ is locally bounded". – Pedro Sánchez Terraf Feb 15 '16 at 18:39
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    @PedroSánchezTerraf the same can be said of paracompactness (where we have a locally finite refinement for every open cover, and the local finiteness sometimes has the same effect as compactness in being able to stitch together local properties to global ones). – Henno Brandsma Feb 15 '16 at 18:51

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