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Let $X_1, X_2, Y$ be topological spaces and let $X_1 \times X_2$ be the topological space obtained by furnishing the Cartesian product set with the product topology. Let $f: X_1 \times X_2 \to Y$ be a given map.

Then f is continuous iff for each $U \in Open(Y)$ and for each $(x_1, x_2) \in X_1 \times X_2$ such that $f(x_1, x_2) \in U$, there exists $(U_1, U_2) \in [Open(X_1) \times Open(X_2)]$ such that $(x_1, x_2) \in U_1 \times U_2$ and $f(U_1 \times U_2) \subseteq U$.

Now, for fixed $x_1$, define a map $f_2^{x_1}:X_2\to Y$ by $x_2 \mapsto f(x_1,x_2)$.

For fixed $x_2$, define a map $f_1^{x_2}:X_1\to Y$ by $x_1 \mapsto f(x_1, x_2)$.

Is there a way to formulate the continuity of $f$ in terms of continuity of $f_1^{x_2}$ for all $x_2$ and continuity of $f_2^{x_1}$ for all $x_1$? If not, what additional assumptions are required? I could prove that if these functions are uniformly continuous (that is, in $x_1$ and $x_2$ respectively) then $f$ is continuous.

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