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

Keywords: mathcal denote property every function colon times mathbb mathbb x continuous every colon mathbb mathcal continuous investigate assumptions theorem luzin which states mathbb mathcal holds mathbb mathcal being class continuous functions mathbb question which topological spaces x holds investigated dalbec here examine mathbb mathcal different families mathcal particular notice mathbb mbox holds where family functions mathbb having continuous directional derivatives allowing infinite values result best possible since mathbb false where family differentiable functions infinite derivatives allowed notice mathcal family graphs functions mathcal subseteq mathcal equivalent property * mathcal every colon times mathbb mathbb upharpoonright continuous every mathcal continuous note mathcal family lines mathbb geq * mathbb mathcal false since there discontinuous linearly continuous functions mathbb direction prove there exists baire class function colon mathbb mathbb * mathbb h holds where stands possible translations subset mathbb times mathbb result best possible since * mathbb h false mathbb notice * mathbb z holds borel subseteq mathbb times mathbb either positive measure second category finally example perfect nowhere dense subseteq mathbb times mathbb measure zero which * mathbb z holds