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 $$A_2$$.

Then $$f$$ is continuous (with respect to the topologies $$\vartheta_1$$ and $$\vartheta_2$$) iff:
 * $$U \in \vartheta_2 \implies f^{-1} \left({U}\right) \in \vartheta_1$$.

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

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

Let $$x \in T_1$$.

Let $$N \subseteq T_2$$ be a neighborhood of $$f \left({x}\right)$$.

Then $$f$$ is continuous at (the point) $$x$$ iff there always exists a neighborhood $$M$$ of $$x$$ such that $$f \left({M}\right) \subseteq N$$.

The general definition for continuous mapping follows from the definition of continuity at a point for all points in the topology.