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 Open Sets
The mapping $f$ is continuous at (the point) $x$ (with respect to the topologies $\tau_1$ and $\tau_2$) iff:
 * For every neighborhood $N$ of $f \left({x}\right)$ in $T_2$, there exists a neighborhood $M$ of $x$ in $T_1$ such that $f \left({M}\right) \subseteq N$.

Definition using Filters
The mapping $f$ is continuous at (the point) $x$ for any filter $\mathcal F$ on $X$ that converges to $x$, the corresponding image filter $f \left({\mathcal F}\right)$ converges to $f \left({x}\right)$.

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