Pointwise Convergence

Pointwise convergence means a sequence of functions converges at each individual point. While natural and easy to define, pointwise convergence does not preserve important properties like continuity or integrability. This motivates the stronger notion of uniform convergence. Understanding the distinction between pointwise and uniform convergence is essential for analysis.

Definition

Definition7.1Pointwise convergence

A sequence of functions $(f_n)$ defined on a set $E$ converges pointwise to a function $f$ if for every $x \in E$ ,

$\lim_{n \to \infty} f_n(x) = f(x).$

That is, for every $x \in E$ and $\epsilon > 0$ , there exists $N$ (depending on $\epsilon$ and $x$ ) such that $|f_n(x) - f(x)| < \epsilon$ for all $n \geq N$ .

RemarkPointwise means point-by-point

Pointwise convergence checks convergence separately at each point. The rate of convergence (the value of $N$ ) can vary wildly from point to point.

Examplef_n(x) = x^n on [0, 1]

Let $f_n(x) = x^n$ on $[0, 1]$ . Then:

For $0 \leq x < 1$ : $f_n(x) = x^n \to 0$ .
For $x = 1$ : $f_n(1) = 1 \to 1$ .

The pointwise limit is $f(x) = 0$ for $x \in [0, 1)$ and $f(1) = 1$ . Note that each $f_n$ is continuous, but the limit $f$ is discontinuous at $x = 1$ . Pointwise convergence does not preserve continuity.

ExampleTriangle wave functions

Define $f_n(x) = n x (1 - x)$ on $[0, 1]$ for $x \in [0, 1/n]$ , and $f_n(x) = (1 - x)$ for $x \in [1/n, 1]$ (after suitable normalization). As $n \to \infty$ , the "peak" moves toward $0$ and grows in height. The pointwise limit is $f(x) = 0$ for all $x \in [0, 1]$ . However, $\int_0^1 f_n(x) \, dx$ may not converge to $\int_0^1 f(x) \, dx$ . Pointwise convergence does not preserve integrals.

Failure to preserve properties

ExamplePointwise limit of derivatives

Let $f_n(x) = \frac{\sin(nx)}{n}$ . Then $f_n(x) \to 0$ pointwise, but $f_n'(x) = \cos(nx) \to \cos(nx)$ does not converge pointwise (it oscillates). Derivatives of $f_n$ do not converge to the derivative of the pointwise limit.

RemarkWhy pointwise convergence is insufficient

Pointwise convergence allows the "speed" of convergence to vary with $x$ . This flexibility means that limits can exhibit pathological behavior: discontinuous limits of continuous functions, non-integrable limits of integrable functions, etc. Uniform convergence fixes this by requiring uniform speed of convergence.

Summary

Pointwise convergence is the weakest notion of convergence for function sequences:

Defined point-by-point: $f_n(x) \to f(x)$ for each $x$ .
Does not preserve continuity, integrability, or differentiability.
Motivates uniform convergence, which does preserve these properties.

See Uniform Convergence for a stronger notion.