\begin{definition}[Definition:Continuous Extension]
Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces.
Let $A, B \subseteq S_1$ be subsets of $S_1$ such that $A \subseteq B$.
Let $f: A \to S_2$ and $g: B \to S_2$ be continuous mappings.
Then $g$ is a '''continuous extension''' of $f$ {{iff}}:
:$\forall s \in A: f \left({s}\right) = g \left({s}\right)$
That is, a '''continuous extension''' of $f$ is a continuous mapping on a superset which agrees with $f$ on the domain of $f$.
Simply, it is a continuous mapping which is an extension.
\end{definition}