Definition:Continuous Mapping (Topology)/Set

Definition
Let $T_1 = \left({A_1, \tau_1}\right)$ and $T_2 = \left({A_2, \tau_2}\right)$ be topological spaces.

Let $f : A_1 \to A_2$ be a mapping from $A_1$ to $A_2$.

Let $S$ be a subset of $A_1$.

The mapping $f$ is continuous on $S$ if $f$ is continuous at every point $x \in S$.