The following property of mappings of two variables is introduced. A function

$f:X\times Y\to \mathbb{R}$ is called quasi-separately continuous at a point

$({x}_{0},{y}_{0})$ if: (1)

${f}_{{x}_{0}}$ – the

${x}_{0}$-section of

$f$ – is continuous at

${y}_{0}$, and (2) for every finite set

$F\subset Y$ and

$\epsilon >0$ there is an open set

$V\subset X$ such that

${x}_{0}\in \text{cl}\left(V\right)$ and

$|f(x,y)-f({x}_{0},y)|<\epsilon $ whenever

$x\in V$ and

$y\in F$.

$f$ is quasi-separately continuous provided if it is quasi-separately continuous at each point

$(x,y)\in X\times Y$. It is shown that if

$X$ is a separable Baire space and

$Y$ is compact then every quasi-separately continuous function

$f:X\times Y\to \mathbb{R}$ has the Namioka property, i.e., there exists a dense

${G}_{\delta}$-set

$D\subset X$ such that

$f$ is jointly continuous at each point of

$D\times Y$. To prove this result the author introduces a new version of a topological game and the notion of quasi-oscillation of a function. The game is a modification of the Saint-Raymond game [

*J. Saint Raymond*, Proc. Am. Math. Soc. 87, 499–504 (1983;

Zbl 0511.54007)].