\begin{definition}[Definition:Continuous Mapping (Topology)/Point/Filters]
Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $f: S_1 \to S_2$ be a mapping from $S_1$ to $S_2$.
Let $x \in S_1$.
The mapping $f$ is '''continuous at (the point) $x$''' {{iff}}:
:for any filter $\FF$ on $T_1$ that converges to $x$, the corresponding image filter $f \sqbrk \FF$ converges to $\map f x$.
\end{definition}