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We study the existence of continuity points for mappings $f: X\times Y\to Z$whose $x$-sections $Y\ni y\to f(x,y)\in Z$ are fragmentable and $y$-sections$X\ni x\to f(x,y)\in Z$ are quasicontinuous, where $X$ is a Baire space and $Z$is a metric space. For the factor $Y$, we consider two infinite "point-picking"games $G_1(y)$ and $G_2(y)$ defined respectively for each $y\in Y$ as follows:In the $n$th inning, Player I gives a dense set $D_n\subset Y$, respectively, adense open set $D_n\subset Y$, then Player II picks a point $y_n\inD_n$; IIwins if $y$ is in the closure of $\{y_n:n\in\mathbb N\}$, otherwise I wins. Itis shown that (i) $f$ is cliquish if II has a winning strategy in $G_1(y)$ forevery $y\in Y$, and (ii) $f$ is quasicontinuous if the $x$-sections of $f$ arecontinuous and the set of $y\in Y$ such that II has a winning strategy in$G_2(y)$ is dense in $Y$. Item (i) extends substantially a result of Debs(1986) and item (ii) indicates that the problem of Talagrand (1985) onseparately continuous maps has a positive answer for a wide class of "small"compact spaces.

Cliquishness and Quasicontinuity of Two Variables Maps