---
tag: 0MVDQEAB60
kind: definition
name: Sequential continuity
msc: [26A15]
defines: 055Z15H69G
uses: [0XKWSSY8P2]
sources:
  - url: https://en.wikipedia.org/wiki/Continuous_function
    locator: Definition in terms of limits of sequences
---

## Definition

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be a function.
The function $f$ is sequentially continuous if the following holds.
For each sequence $(x_n)$ in $D$ that [[converges|0XKWSSY8P2]] to a point $a \in D$, the sequence $(f(x_n))$ converges to $f(a)$.

## Remarks

For real functions, [[sequential continuity is the same as continuity|0PAQM3H57X]].
The definition is useful to prove that a function is not continuous. It is enough to find one sequence $x_n \to a$ with $f(x_n) \not\to f(a)$.
Mathlib defines it for topological spaces, as `SeqContinuous`.

## Explanations

### L1 human

A sequentially continuous function preserves limits of sequences.
The function of the limit is the limit of the function values.