Uniform Convergence

• categories: Real Analysis, Definition

Uniform convergence of a sequence of functions means that the functions converge to a limit function at the same rate across the entire 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 uniformly to a function $f(x)$ on $D$ if, for every $ϵ>0$, there exists an integer $N$ such that for all $n\ge N$ and for all $x\in D$:

$$|f_n(x)-f(x)|<ϵ$$

The key point here is that $N$ does not depend on $x$; it is uniform for all $x\in D$.

Formal Statement

The sequence of functions $\{f_n(x)\}$ converges uniformly to $f(x)$ if:

$$\underset{n\to \infty }{\lim }\underset{x\in D}{\sup }|f_n(x)-f(x)|=0$$

This means that the maximum difference between $f_n(x)$ and $f(x)$ over the entire domain $D$ can be made arbitrarily small for sufficiently large $n$, and this holds uniformly across all $x\in D$.

Intuition

• In uniform convergence, the speed of convergence is the same across the entire domain. This contrasts with pointwise convergence, where the rate of convergence can vary depending on the point $x$.
• Uniform convergence guarantees that the function sequence behaves consistently over the whole domain, making it stronger than pointwise convergence.

Example

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

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

For $x\in [0,1)$, $f_n(x)\to 0$ as $n\to \infty$, but for $x=1$, $f_n(1)=1$ for all $n$. The pointwise limit is:

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

However, $\{f_n(x)\}$ does not converge uniformly to $f(x)$ because the rate of convergence varies depending on $x$. Specifically, near $x=1$, the convergence is slower, and no single $N$ can ensure $|f_n(x)-f(x)|<ϵ$ for all $x\in [0,1]$.

Comparison to Pointwise Convergence

• In pointwise convergence, convergence happens for each point individually, and the rate of convergence can vary with $x$.
• In uniform convergence, the convergence occurs at the same rate over the entire domain, providing stronger guarantees about the behavior of the sequence.