Pointwise Convergence

• categories: Real Analysis, Definition

Pointwise convergence describes the convergence of a sequence of functions at each individual point in the domain.

Definition

Let $\{f_n(x)\}$ be a sequence of functions defined on a set $D\subseteq \mathbb{R}$. The sequence $\{f_n(x)\}$ converges pointwise to a function $f(x)$ on $D$ if, for every $x\in D$:

$$\underset{n\to \infty }{\lim }{f}_n(x)=f(x)$$

In other words, for each fixed $x\in D$, the values $f_n(x)$ approach $f(x)$ as $n\to \infty$.

Formal Statement

Given a sequence of functions $\{f_n(x)\}$ defined on $D$, the pointwise limit of $\{f_n(x)\}$ is the function $f(x)$ such that, for every $x\in D$ and for every $ϵ>0$, there exists an integer $N=N(x,ϵ)$ such that for all $n\ge N$:

$$|f_n(x)-f(x)|<ϵ$$

This means that for each point $x$, after a sufficiently large number of steps, $f_n(x)$ gets arbitrarily close to $f(x)$.

Intuition

• In pointwise convergence, the convergence happens for each individual point $x$ separately.
• It does not require the same rate of convergence for all points; the function may converge faster at some points than others.

Example

Consider the sequence of functions $\{f_n(x)\}$ on the interval $[0,1]$ given by:

$$f_n(x)=x^n$$

As $n\to \infty$:

• For $x=1$, $f_n(1)=1$ for all $n$.
• For $x\in [0,1)$, $f_n(x)\to 0$ as $n\to \infty$.

Hence, the pointwise limit function is:

$$f(x)=\{\begin{array}{ll}0,& \text{if }0\le x<1\\ 1,& \text{if }x=1\end{array}$$

Comparison to Uniform Convergence

In uniform convergence, the entire sequence of functions converges to $f(x)$ at the same rate across the domain, meaning that the speed of convergence does not depend on $x$. Pointwise convergence allows for different rates of convergence at different points, which can lead to discontinuities in the limit function.