Continuity at a point of a topological space

Definition

Let $X$ and $Y$ be topological spaces, let $f \colon X \to Y$ be a function and let $a \in X$. The function $f$ is continuous at $a$ when for each neighbourhood $V$ of $f(a)$, the preimage $f^{-1}(V)$ is a neighbourhood of $a$.

Remarks

Equivalently, for each neighbourhood $V$ of $f(a)$ there is a neighbourhood $U$ of $a$ with $f(U) \subseteq V$. A function is a continuous map if and only if it is continuous at each point of $X$. For metric spaces, continuity at $a$ is the same as the epsilon-delta condition at a.

Explanations

L1 human

Continuity at $a$ means that you can keep $f(x)$ as close to $f(a)$ as you want, if you keep $x$ close enough to $a$. Neighbourhoods replace the distances of the metric case.