# Definition:Continuous Mapping (Topology)/Point

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## 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 $\tau_2$) if and only if:

- For every neighborhood $N$ of $\map f x$ in $T_2$, there exists a neighborhood $M$ of $x$ in $T_1$ such that $f \sqbrk M \subseteq N$.

### Definition using Filters

The mapping $f$ is **continuous at (the point) $x$** if and only if for any filter $\mathcal F$ on $T_1$ 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
- Definition:Discontinuous at Point of Topological Space

- Results about
**continuous mappings**can be found here.