Baire classes and separately continuous functions

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Problem III.1.6 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section III.1 "Continuity and Discontinuities of Functions"

Let f0f_{0} be the Dirichlet function:

f0(x)={1, for x∈Q,0, for x∈R\Q. f_{0}(x)=\begin{cases} 1, & \text{ for } x \in \mathbb {Q}, \\ 0, & \text{ for } x \in \mathbb {R} \backslash \mathbb {Q} . \end{cases}
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(a)

Prove that for any x∈Rx \in \mathbb {R}

f0(x)=lim⁡m→∞lim⁡n→∞cos⁡2n(2πxm!) f_{0}(x)=\lim _{m \rightarrow \infty } \lim _{n \rightarrow \infty } \cos ^{2 n}(2 \pi x m!)
(b)

Does there exist a sequence of functions {fn}\left\{ f_{n}\right\} continuous on R\mathbb {R} such that f0(x)=lim⁡n→∞fn(x)f_{0}(x)=\lim_{n \rightarrow \infty } f_{n}(x) for any x∈Rx \in \mathbb {R}?

Problem III.1.27 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section III.1 "Continuity and Discontinuities of Functions"

Let the function ff be defined and separately continuous on the square Q=[0,1]2Q=\left[0,1\right]^{2}. Prove that there exists a sequence fnf_{n} of functions continuous on QQ such that fn(x,y)→f(x,y)f_{n}(x, y) \rightarrow f(x, y) for all x,y∈[0,1]x, y \in [0,1] (i.e., ff is a function of the first Baire class on QQ).

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Problem III.1.28 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section III.1 "Continuity and Discontinuities of Functions"

Let the function ff be defined and separately continuous on R2\mathbb {R}^{2}. Prove that if ff vanishes on an everywhere dense set (i.e., a set whose closure coincides with R2\mathbb {R}^{2}), then f≡0f \equiv 0.

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References

[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.