Definition:Continuous Mapping (Topology)/Point

Definition
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 $x \in S_1$.

Definition using Filters
If necessary, we can say that $f$ is $\left({\tau_1, \tau_2}\right)$-continuous at $x$.

Also see

 * Equivalence of Definitions of Continuous Mapping between Topological Spaces at Point