---
tag: 056FSKG32J
kind: definition
name: Pointwise convergence
msc: [40A30]
defines: 0GFGK40582
uses: [0XKWSSY8P2]
sources:
  - url: https://en.wikipedia.org/wiki/Pointwise_convergence
    locator: Definition
---

## Definition

Let $S$ be a set, and let $f_n \colon S \to \mathbb{R}$ for $n \ge 1$ and $f \colon S \to \mathbb{R}$ be functions.
The sequence $(f_n)$ converges pointwise to $f$ on $S$ if $\lim_{n \to \infty} f_n(x) = f(x)$ for every $x \in S$.
That is, for every $x \in S$ and every $\varepsilon > 0$ there is $N$ with $|f_n(x) - f(x)| < \varepsilon$ for all $n \ge N$.

## Remarks

Here $N$ may depend on both $\varepsilon$ and $x$.
The pointwise limit is unique, since [[limits|0XKWSSY8P2]] of real sequences are unique.
A pointwise limit of continuous functions need not be continuous: see [[the sequence $x^n$|0YSRR1JTKZ]]. Mathlib expresses pointwise convergence as convergence in the product topology, through `tendsto_pi_nhds`.

## Explanations

### L1 human

Pointwise convergence means that at each single point, the values $f_n(x)$ approach $f(x)$.
Different points may converge at very different speeds.