\begin{definition}[Definition:Continuous Mapping (Topology)/Set]
Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces.
Let $f: S_1 \to S_2$ be a mapping from $S_1$ to $S_2$.
Let $S$ be a subset of $S_1$.
The mapping $f$ is '''continuous on $S$''' {{iff}} $f$ is continuous at every point $x \in S$.
\end{definition}