Restricted continuity and a theorem of Luzin

Krzysztof Chris Ciesielski¹ ; Joseph Rosenblatt²

¹ Department of Mathematics West Virginia University Morgantown, WV 26506-6310, U.S.A. and Department of Radiology MIPG University of Pennsylvania 4th Floor, Blockley Hall 423 Guardian Drive Philadelphia, PA 19104-6021, U.S.A.
² Department of Mathematics University of Illinois Urbana, IL 61801, U.S.A.

Colloquium Mathematicum, Tome 135 (2014) no. 2, pp. 211-225.

Résumé

Let $P(X,\mathcal F)$ denote the property: For every function $f\colon X\times \mathbb R\to\mathbb R$, if $f(x,h(x))$ is continuous for every $h\colon X\to\mathbb R$ from $\mathcal F$, then $f$ is continuous. We investigate the assumptions of a theorem of Luzin, which states that $P(\mathbb R,\mathcal F)$ holds for $X=\mathbb R$ and $\mathcal F$ being the class $C(X)$ of all continuous functions from $X$ to $\mathbb R$. The question for which topological spaces $P(X,C(X))$ holds was investigated by Dalbec. Here, we examine $P(\mathbb R^n,\mathcal F)$ for different families $\mathcal F$. In particular, we notice that $P(\mathbb R^n,``C^1\mbox{''})$ holds, where “$C^1$” is the family of all functions in $C(\mathbb R^n)$ having continuous directional derivatives allowing infinite values; and this result is the best possible, since $P(\mathbb R^n,D^1)$ is false, where $D^1$ is the family of all differentiable functions (no infinite derivatives allowed). We notice that if $\mathcal D$ is the family of the graphs of functions from $\mathcal F\subseteq C(X)$, then $P(X,\mathcal F)$ is equivalent to the property $P^*(X,\mathcal D)$: For every $f\colon X\times \mathbb R\to\mathbb R$, if $f\upharpoonright D$ is continuous for every $D\in\mathcal D$, then $f$ is continuous. Note that if $\mathcal D$ is the family of all lines in $\mathbb R^n$, then, for $n\geq 2$, $P^*(\mathbb R^n,\mathcal D)$ is false, since there are discontinuous linearly continuous functions on $\mathbb R^n$. In this direction, we prove that there exists a Baire class 1 function $h\colon \mathbb R^n\to\mathbb R$ such that $P^*(\mathbb R^n,T(h))$ holds, where $T(H)$ stands for all possible translations of $H\subset \mathbb R^n\times\mathbb R$; and this result is the best possible, since $P^*(\mathbb R^n,T(h))$ is false for any $h\in C(\mathbb R^n)$. We also notice that $P^*(\mathbb R^n,T(Z))$ holds for any Borel $Z\subseteq\mathbb R^n\times\mathbb R$ either of positive measure or of second category. Finally, we give an example of a perfect nowhere dense $Z\subseteq\mathbb R^n\times\mathbb R$ of measure zero for which $P^*(\mathbb R^n,T(Z))$ holds.

Détail

DOI : 10.4064/cm135-2-5

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