Separately continuous map

Definition

Suppose $A,B,C$ are topological spaces. Suppose $f:A\times B\to C$ is a set map. We say that $f$ is separately continuous if it satisfies the following two conditions:

1. For every $a\in A$, the map $b\mapsto f(a,b)$ is a continuous map from $B$ to $C$.
2. For every $b\in B$, the map $a\mapsto f(a,b)$ is a continuous map from $A$ to $C$.

Equivalently, $f$ is separately continuous if it is continuous as a map from $A\times B$ to $C$ where $A\times B$ is given the slice topology.