Definition:Continuous Mapping (Topology)/Point

Definition
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 $x \in S_1$.

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

Also see

 * Equivalence of Definitions of Continuous Mapping between Topological Spaces at Point
 * Definition:Discontinuous at Point of Topological Space