Definition:Continuous Mapping (Topology)

Definition
Let $T_1 = \left({A_1, \vartheta_1}\right)$ and $T_2 = \left({A_2, \vartheta_2}\right)$ be topological spaces.

Let $f : A_1 \to A_2$ be a mapping from $A_1$ to $A_2$.

Continuous at a Point
Let $x \in A_1$.

Let $N \subseteq A_2$ be a neighborhood of $f \left({x}\right)$.

Then $f$ is continuous at (the point) $x$ (with respect to the topologies $\vartheta_1$ and $\vartheta_2$) iff there always exists a neighborhood $M$ of $x$ such that $f \left({M}\right) \subseteq N$. Here, $f \left({M}\right)$ denotes the image of $M$ under $f$.

If necessary, we can say that $f$ is $\left({\vartheta_1, \vartheta_2}\right)$-continuous at $x$.

Continuous on a Set
Let $S$ be a subset of $A_1$.

The mapping $f$ is continuous on $S$ if $f$ is continuous at every point $x \in S$.

Continuous Everywhere
The mapping $f$ is continuous everywhere (or simply continuous) if $f$ is continuous at every point $x \in A_1$.

Alternatively, one could specify that:
 * $U \in \vartheta_2 \implies f^{-1} \left({U}\right) \in \vartheta_1$

where $f^{-1} \left({U}\right)$ denotes the preimage of $U$ under $f$.

That these conditions are equivalent is proven in Equivalence of Definitions of Continuous Mapping (Topology).