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We generalize a result of Luzin about real functions of real variables -that F(x, y) is jointly continuous if and only if F(x, f(x)) is continuous for every continuous f-to functions among certain classes of topological spaces. In [3, pp. 173-176] Luzin states and proves that a function F: [a, b] x [c, d] -* R is continuous if and only if its restriction to the graph of each continuous function f: [a, b] -* [c, d] and g: [c, d] -* [a, b] is continuous. In this theorem, necessity is easy to show even when [a, b], [c, d], and R are replaced by arbitrary topological spaces, so we will concern ourselves with proving sufficiency. An examination of Luzin's sufficiency proof shows that he uses only the hypothesis that F is continuous when restricted to the graph of each continuous function f: [a, b] -* [c, d], or equivalently that the function Ff: [a, b] -* R is continuous for each such f, where Ff(x) = F(x, f(x)). Our generalizations will use a form of this hypothesis (Theorem 2) or a slightly stronger one (Theorem 1). We recall some definitions and results from [1] for easy reference. A space is called Frechet if the closure of a set includes precisely the limits of all sequences in the set. A space is called sequential if a set is closed precisely when it includes the limits of all sequences included in the set. We will use many other topological terms defined in [1, 4] without explicit mention. Lemma 1 [1, p. 78]. If a sequence converges to a point, all of its subsequences also converge to that point. Lemma 2 [ 1, p. 11 1]. The product of two first countable spaces is first countable. Lemma 3 [1, p. 78]. A first countable space is Frechet and a Frechet space is sequential. Lemma 4 [1, p. 271]. The product of two sequential spaces, one of which is sequentially compact, is sequential. Received by the editors April 8, 1991 and, in revised form, October 16, 1991. 1991 Mathematics Subject Classification. Primary 54C05, 54B10; Secondary 26B05. The author wishes to thank the referee for many helpful suggestions. This paper is based on the author's Master's thesis at Youngstown State University. i 1993 American Mathematical Society 0002-9939/93 $1.00 + $.25 per page

When does restricted continuity on continuous function graphs imply joint continuity?