Definition:Continuous Mapping (Topology)/Point

Definition
Let $T_1 = \left({A_1, \tau_1}\right)$ and $T_2 = \left({A_2, \tau_2}\right)$ be topological spaces.

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

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 $\tau_1$ and $\tau_2$) iff there always exists a neighborhood $M$ of $x$ such that $f \left({M}\right) \subseteq N$.

If necessary, we can say that $f$ is $\left({\tau_1, \tau_2}\right)$-continuous at $x$.