Functions nice in each variable separately

Consider the following question. If {f:{\mathbb R}^2\rightarrow {\mathbb R}} 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

\displaystyle  f(x,y):= \frac {xy}{x^2+y^2}

Defining {f(0,0)=0}, 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 {f:{\mathbb C}^{n_1}\times {\mathbb C}^{n_2}\rightarrow {\mathbb C}} is holomorphic in each variable separately. Then {f} is holomorphic on {{\mathbb C}^{n_1+n_2}}.

Below the fold, I will state some more results of this type and prove some of them.

Definition 2 A function {f:{\mathbb X}\rightarrow {\mathbb R}} is of Baire class {1} if it is the pointwise limit of continuous functions.

Remark 1 One can define functions of Baire class {n} as pointwise limits of functions of Baire class {n-1}, and this definition extends to any ordinal {\omega}.

Theorem 3 (Baire) Suppose {f:{\mathbb R}\times {\mathbb R}\rightarrow {\mathbb R}} is continuous in each variable separately. Then {f} is of Baire class {1}.

Proof: Take vertical lines in the plane, spaced at distance {\frac 1n}. Define {f_n(x,y)} to be the linear interpolation of {f(x,y)} restricted to each line. Because {f} is also continuous in the horizontal direction, these functions converge pointwise to {f}. \Box

Remark 2 Another theorem of Baire states that a function is of Baire class {1} 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 {F_\sigma} sets.

Recall that continuous functions are defined as those for which preimages of open sets are open. Note also that open sets are {F_\sigma}.

Theorem 4 A function {f:X\rightarrow {\mathbb R}} is of Baire class {1} if and only if the preimage of any open set is {F_\sigma}.

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.

Lemma 5

  • (i) The indicator function of a closed set is of Baire class {1}.
  • (ii) The indicator function of a set which is {F_\sigma} and {G_\delta} is of Baire class {1}.
  • 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 {F_\sigma}). \Box

    Lemma 6 Suppose {X=\cup_{i=1}^{N} A_i} is a finite union of {F_\sigma} sets. Then there exist {B_i\subset A_i} which are simultaneously {F_\sigma} and {G_\delta}, disjoint, and such that

    \displaystyle  X=\coprod_{i=1}^N B_i

    Proof: Write {A_i} as a countable union of closed sets. There is a natural way to iteratively construct disjoint candidates {B_i}, and they will have the required property because countable union of {F_\sigma} is {F_\sigma}, and countable intersection of {G_\delta} is {G_\delta}. \Box

    We can now return to the proof of Theorem 4.

    Proof: In one direction, suppose {f} is the pointwise limit of {f_n}. Then we have

    \displaystyle  \{x|f(x)<r\}=\bigcup_{\epsilon,N}\bigcap_{n\geq N} \{x|f_n(x)\leq r-\epsilon\}

    In the other direction, assume the preimage of an open is always {F_\sigma}. We also assume that the image of {f} is in some open interval {I} (this is allowed, since {I} is homeomorphic to {{\mathbb R}}). Cover the interval {I} by small open sets of size {\frac{|I|}N}. Their preimages are {F_\sigma}, and by Lemma 6 we can replace them by disjoint, simultaneously {F_\sigma} and {G_\delta} sets. Take indicator functions of these sets (with weights corresponding to the image in {I}). These will be of Baire class {1} by Lemma 5 and as {N\rightarrow \infty}, they will uniformly converge to {f}. The uniform limit of Baire class {1} functions is still of Baire class {1}, so the
    claim is proved. \Box

    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 {f:{\mathbb R}^2\rightarrow {\mathbb R}} is real-analytic in each variable separately. Then there exists a closed set {F\subset {\mathbb R}^2} such that {f} is real-analytic off this set. Moreover, the projection of {F} 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 {x_1,x_2,\cdots} and a point {z}, we can define the first return sequence {R_i(z)} of {z} as the subsequence of {x_i} which comes (strictly) closer than all the previous elements of the sequence. If the value of {f(z)} is equal to the limit of {f} evaluated at {R_i(z)}, then we say that {f} can be “first-return recovered” from the sequence.

    A non-obvious theorem is that if {f} is of Baire class {1} if and only if there exists a sequence which first-return recovers it.

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