Let $(X,\mathscr{T}_{X})$ and $(Y,\mathscr{T}_{Y})$ be topological spaces, and let $f:X\to Y$ be a map. 1. The map $f$ is continuous at $x_{0}$ for $x_{0}\in X$ if, for each neighbourhood $\mathcal{V}$ of $f(x_{0})$, there exists a neighbourhood $\mathcal{U}$ of $x_{0}$ such that $$f(\mathcal{U})\subset \mathcal{V}.$$ 2. If $f$ is continuous at each $x\in X$ then it is continuous. One may verify that $f$ is continuous if and only if $$\forall \mathcal{O}\in \mathscr{T}_{Y}: f^{-1}(\mathcal{O})\in \mathscr{T}_{X}$$ 3. 4.