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Suppose $X$ is a metric space. A real valued function $f$ defined on a metric space $X$ is said to be of first baire class, if $f$ is a pointwise, limit of a sequence of continuous functions on $X$.

A famous well know theorem asserts that if $f$ is in the first baire class, then the set of points of discontinuity is of the first category.

Using this definition, can we characterize for what Metric Spaces, will the points of continuity of a real valued function, be dense in itself.

Ok, since we are talking about metric spaces, lets consider the 2 famous ones, complete and Compact. A compact metric space, is "Complete and Totally Bounded" by definition, so lets not look into it. So lets look at complete metric spaces.

So the question basically, is if, $X$ is a complete metric space, and $f$ is as above(1st Baire class), then is the points of continuity of $f$ dense in $X$.

There are some exercises related, but i didn't take it from here. So dont misunderstand me. http://books.google.com/books?id=CoSIe7h5mTsC&pg=PA166&dq=f+is+of+first+baire+class,+points+of+continuity

J. M. ain't a mathematician
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  • I am a bit confused by the statement of the question. The warmup to the question features Baire class one functions, but the question itself does not. – Pete L. Clark Sep 16 '10 at 03:44
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    Is that a quote from some homework assignment or some book? – Asaf Karagila Sep 16 '10 at 06:05
  • @Asaf: No! Its not an homework! –  Sep 16 '10 at 08:48
  • @Asaf: While i was reading Baire category, theorem, i got this in my mind. Generally BCT talks abt the points of discontinuity, i asked why not ask something about the points of continuity of a function. –  Sep 16 '10 at 08:56

1 Answers1

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Yes. The Baire category theorem asserts that in a complete metric space, a set of first category not only is not the whole space, but actually has dense complement.

Nate Eldredge
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