Pointwise convergence

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 of real sequences are unique. A pointwise limit of continuous functions need not be continuous: see the sequence $x^n$. 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.