# Definition:Continuous Mapping (Topology)/Point

< Definition:Continuous Mapping (Topology)(Redirected from Definition:Continuous Mapping at Point (Topology))

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## 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 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 open set $U_2$ of $T_2$ such that $\map f x \in U_2$, there exists an open set $U_1$ of $T_1$ such that $x \in U_1$ and $f \sqbrk {U_1} \subseteq U_2$.

### Definition using Neighborhoods

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 $\FF$ on $T_1$ that converges to $x$, the corresponding image filter $f \sqbrk \FF$ converges to $\map f x$.

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

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