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Let $E_1$ be a closed measure zero subset of the unit circle $C$. Moreover, assume that there exists a set $E_2\subset C$, such that $E_1\subset E_2$ and a bounded analytic function $f$ defined on the open unit disc $D$ $f:D\rightarrow\mathbb{C}$, satisfying the following:

a) $f$ fails to have radial limits exactly on $E_1$

b) $f$ has unrestricted limits exactly on $C\setminus E_2$. (i.e $f$ can be extended continuously exactly on $C\setminus E_2$).

Then it follows that the function $f^*$ defined on $C$ to take the values of the radial limits of $f$ when they exist, is well defined on the set $C\setminus E_1$ and it is of the Baire first class. My question is then the following:

Is there any Baire's theorem that guarantees that the set of discontinuities of $f^*$ (i.e the set $E_2\setminus E_1\subset C\setminus E_1$) is meager on $C\setminus E_1$.

I know that there exists a theorem like that for Banach spaces, however in my case $C\setminus E_1$ is not complete.

Thanks in advance for any help.

TheGeometer
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  • You may want to rephrase your question so that the answer is not automatically YES by a standard result of Baire one functions. It doesn't have to be precisely stated, but something that isn't clearly a misrepresentation. Also, you might want to state explicitly, when $E_2$ is introduced, that $E_2$ is a subset of $C.$ Also, is $f$ real-valued or complex-valued? And perhaps define "unrestricted limit" -- does this mean that at each $x \in C\setminus E_2$ the cluster set at $x$ (presumably unrestricted cluster set, and not tangential, radial, etc.) (continued) – Dave L. Renfro Jun 01 '22 at 19:07
  • is an unbounded subset of the co-domain, or does this mean that at each $x \in C\setminus E_2$ the limit exists in the usual sense (i.e. no restriction on approach), or something else (e.g. some kind of non-uniformity of limiting behavior as the point $x$ is varied over $x \in C\setminus E_2)$? – Dave L. Renfro Jun 01 '22 at 19:14
  • @DaveL.Renfro I edited the question now. Moreover, I think the standard result you are referring is about complete metric spaces which is not the case here – TheGeometer Jun 01 '22 at 19:50
  • I don't know much about this topic, but because I bump into it a lot due to my interests, maybe the following will help in a literature search. See the names and references in the paragraph where I discuss complex-valued functions in my answer to Name for multi-valued analogue of a limit. Besides the names listed there, three others especially prominent in this area are John Emerson McMillan and Valerian Ivanovich Gavrilov and Hidenobu Yoshida. – Dave L. Renfro Jun 01 '22 at 20:15
  • @DaveL.Renfro Thank you I will look into it – TheGeometer Jun 01 '22 at 20:19

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