Consider the following question. If is continuous in each variable separately, need it be continuous in both? Although Cauchy (incorrectly) said it should, note it is not even obvious that it should be measurable.
A classical example is
Defining , this function is real-analytic in each variable separately, but it is not even continuous at the origin.
By contrast, in the complex-analytic situation we have the following result.
Theorem 1 (Hartogs) Suppose is holomorphic in each variable separately. Then is holomorphic on .
Below the fold, I will state some more results of this type and prove some of them.
Definition 2 A function is of Baire class if it is the pointwise limit of continuous functions.
Remark 1 One can define functions of Baire class as pointwise limits of functions of Baire class , and this definition extends to any ordinal .
Theorem 3 (Baire) Suppose is continuous in each variable separately. Then is of Baire class .
Proof: Take vertical lines in the plane, spaced at distance . Define to be the linear interpolation of restricted to each line. Because is also continuous in the horizontal direction, these functions converge pointwise to .
Remark 2 Another theorem of Baire states that a function is of Baire class if and only if it is continuous everywhere, except possibly on a meager set. Recall that a set is meager if if it is the countable union of nowhere dense sets. Such sets are always contained in meager sets.
Recall that continuous functions are defined as those for which preimages of open sets are open. Note also that open sets are .
Remark 3 One can define further hierarchies of functions extending the above theorem, and related to the definition of functions of higher Baire class.
Before proving the theorem, we need some preliminaries.
(i) The indicator function of a closed set is of Baire class . (ii) The indicator function of a set which is and is of Baire class .
Proof: Part (i) follows from the Tietze extension theorem. Similarly, part (ii) follows by the same type of argument, but applied to the set and its complement (both are ).
Proof: Write as a countable union of closed sets. There is a natural way to iteratively construct disjoint candidates , and they will have the required property because countable union of is , and countable intersection of is .
We can now return to the proof of Theorem 4.
Proof: In one direction, suppose is the pointwise limit of . Then we have
In the other direction, assume the preimage of an open is always . We also assume that the image of is in some open interval (this is allowed, since is homeomorphic to ). Cover the interval by small open sets of size . Their preimages are , and by Lemma 6 we can replace them by disjoint, simultaneously and sets. Take indicator functions of these sets (with weights corresponding to the image in ). These will be of Baire class by Lemma 5 and as , they will uniformly converge to . The uniform limit of Baire class functions is still of Baire class , so the
claim is proved.
I would like to end with a couple of related remarks which I found interesting. The first result is due to Saint-Raymond.
Theorem 7 Suppose is real-analytic in each variable separately. Then there exists a closed set such that is real-analytic off this set. Moreover, the projection of to each coordinate axis has zero logarithmic capacity, in particular the projection has Lebesgue measure zero.
Finally, there is a whole direction of research (see related references) concerned with the following type of situation. If we know the values of a continuous function at a dense set of points, then we know the function.
Now, given a dense sequence of points and a point , we can define the first return sequence of as the subsequence of which comes (strictly) closer than all the previous elements of the sequence. If the value of is equal to the limit of evaluated at , then we say that can be “first-return recovered” from the sequence.
A non-obvious theorem is that if is of Baire class if and only if there exists a sequence which first-return recovers it.