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 $f(x)=|x|.$

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$.

Prerequisites¶

• Function
• Limit of a Sequence

Related Concepts¶

• Uniform Convergence
• Convergence in Probability

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