Definition 1

A mapping $f$ of a topological space $X$ into a topological space $X'$ is said to be continuous at a point $x_0 \in X$ if, given any neighbourhood $V'$ of $f(x_0)$ in $X'$, there is a neighbourhood $V$ of $x_0$ in $X$ such that the relation $x \in V$ implies $f(x) \in V'$.

Definition 1 may be restated in the following more intuitive form: to say that $f$ is continuous at the point $x_0$ means that $f(x)$ is as near as we please to $f(x_0)$ whenever $x$ is sufficiently near $x_0$.

The relation "for each $x \in V, f(x) \in V'$" is equivalent to $f(V) \subset V'$ or again to $V \subset \overline{f}^{-1}(V')$; in view of the neighbourhood axiom ($V_I$), we see that Definition 1 is equivalent to the following: $f : X \to X'$ is said to be continuous at the point $x_0$ if, for each neighbourhood $V'$ of $f(x_0)$ in $X'$, $\overline{f}^{-1}(V')$ is a neighbourhood of $x_0$ in $X$. Moreover, it is sufficient that $\overline{f}^{-1}(V')$ is a neighbourhood of $x_0$ for each neighbourhood $V'$ belonging to a fundamental system of neighbourhoods of $f(x_0)$ in $X'$ ($§ 1$, no. 3).