Pointwise Convergence¶

Definition¶

Let $(f_n)_{n=1}^{\infty}$ be a sequence of functions $f_n\colon X\to \mathbb{R}$ and $f\colon X \rightarrow \mathbb{R}$ be another function. We say that $(f_n)_{n=1}^{\infty}$ converges pointwise to $f$ on $X$ and write $f_n\to f\text{,}$ if for every $x \in X\text{,}$ $\displaystyle \lim _{n \rightarrow \infty} f_n(x) = f(x)\text{.}$

Each $f_n$ is a Function, so fixing $x$ gives an ordinary numerical sequence $f_n(x)$ . Its Limit of a Sequence is $f(x)$ ; the required index may depend on this fixed $x$ .

Interpretation¶

Pointwise convergence means that for each fixed point $x\in X$ $X$ , the numbers $f_n(x)$ form an ordinary sequence that tends to $f(x)$ as $n\to\infty$ $\infty$ . The functions are compared with the limit one point at a time, and nothing is required about how the points relate to each other.

Written out with quantifiers: for every $x\in X$ and every $\varepsilon>0$ there is an $N$ such that $|f_n(x)-f(x)|<\varepsilon$ for all $n>N$ $N$ . The index $N$ may depend on both $\varepsilon$ and $x$ . If one $N$ can be chosen for all $x$ at once, the convergence is uniform; the two notions differ only in this order of quantifiers.

Because the required $N$ may grow without bound as $x$ moves, properties of the $f_n$ need not survive in the limit, as the Key Results state. The functions $x^n$ on $[0,1]$ are continuous, but their pointwise limit is $0$ on $[0,1)$ and $1$ at $x=1$ $1$ , so it is not continuous. Integrals are not preserved either: on $[0,1]$ , let $f_n=n$ on $(0,\tfrac{1}{n})$ and $f_n=0$ elsewhere. Then $f_n(x)\to0$ at every $x$ , while $\int_0^1 f_n(x)\,\mathrm{d}x=1$ for every $n$ .

A common slip is to let the point move with $n$ . Pointwise convergence fixes $x$ first and only then lets $n\to\infty$ $\infty$ . For $x^n$ , the values $(1-\tfrac{1}{n})^n$ tend to $\mathrm{e}^{-1}$ , not to $0$ ; this does not contradict pointwise convergence to $0$ on $[0,1)$ , but it does show that the convergence is not uniform.

The same idea returns in probability: almost-sure convergence of random variables is pointwise convergence on the sample space outside a set of probability zero.

Key Results¶

Pointwise convergence guarantees that the limit exists at each individual point.
Pointwise convergence does not preserve boundedness, limits/continuity, differentiability, or integrals.

Examples¶

$f_n(x) = \dfrac{n}{nx + 1}$ converges pointwise to $f(x)=1/x$ on $(0,+\infty)\text{.}$
$f_n(x)=\dfrac{x^2}{\sqrt{x^2+1 / n}}$ converges pointwise to $f(x)=|x|.$

Functions x squared over the square root of x squared plus 1/n for n = 1, 2, 4, 8 and 16, approaching the graph of the absolute value; at x = -1 the values rise towards 1.

For each fixed $x$ , $f_n(x)\to|x|$ $∣ x ∣$ ; at $x=-1$ the values $f_n(-1)$ (dots) increase to $f(-1)=1$ $1$ .

Point at a value $x$ : the numbers $f_n(x)$ (right) enter the $\varepsilon$ -band from some $N(x)$ on. For $x^n$ that $N(x)$ grows without bound as $x\to1$ $1$ .

Prerequisites¶

Sources¶

Math III: Analysis · Lecture 1

Aliases: converges pointwise, pointwise limit Symbols: element of, epsilon, infinity, integral, real numbers, square root

Last updated: 20 October 2026 03:02