---
tag: 07N7W5C29X
kind: definition
name: Continuity at a point of a topological space
msc: [54C05]
defines: 0KRMJRDRZT
uses: [0CNQXVBHCX, 09CJ00H3NW, 0FGCFY1AZX]
sources:
  - url: https://en.wikipedia.org/wiki/Continuous_function
    locator: Continuity at a point
---

## Definition

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]], 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|09CJ00H3NW]] $V$ of $f(a)$, the [[preimage|0FGCFY1AZX]] $f^{-1}(V)$ 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|0SY9ECQ19Q]] 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|0EKGKY5TGH]].

## 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.