Pointwise vs Uniform Convergence

It is often useful to take the limit of a sequence of *functions*, with the codomain being a subset of $\mathbb{R}$(or more generally, a metric space). Although it may seem very obvious but there are some subtleties on how to define the convergence of such a sequence.

There are two approaches:

• Pointwise:

A sequence of functions $f_{n}$ with $f_{n}:A\to \mathbb{R}$ where $A$ is an arbitrary set, is said to converge pointwise to $f:A\to \mathbb{R}$ iff for every $x$ in $A$ and every $\epsilon>0$ there exists $N \in \mathbb{N}$ such that $n>N$ implies $|f_{n}-f|<\epsilon)$

• Uniform:

A sequence of A sequence of functions $f_{n}$ with $f_{n}:A\to \mathbb{R}$ where $A$ is an arbitrary set, is said to converge uniformly to $f:A\to \mathbb{R}$ iff for every $x$ in $A$ there exists $N \in \mathbb{N}$ such that for every $\epsilon>0$ $n>N$ implies $|f_{n}-f|<\epsilon)$.

It can take a little while to dismantle the definitions and understand the difference between the two, but the core difference between the two concepts is that in pointwise convergence we only require that for each point,$x\in A$, the $f_{n}(x)$’s are eventually as close as we like to $f(x)$, while in uniform convergence we require that the $f_{n}$’s for all points in the domain, would all be eventually as close as we like to $f$.

It is helpful to consider a sequence of of functions which converges pointwise but not uniformly.

Consider $A=[0,1)$ and $f_{n}=x^n$, then $f_{n}$ converges pointwise to zero but not uniformly,

Every point in $[0,1)$ eventually goes to $0$, for instance focus on the grey point, it has an $x$ value very close to $1$, but it still goes to $0$ eventually, the same is true for any point, however together the sequence of functions is never completely less than even $0.5$, and so while the sequence converges pointwise to $0$ (as depicted by the grey dot) it does not converge uniformly(since it’s always ‘stuck’ to $1$).

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Created on May 19, 2023 By vivaandaga

Uniform Convergence Quiz

1 / 5

Do pointwise limits of continuous functions need to be continuous?

True

False

2 / 5

Which of the following functions is uniformly convergent?

$e^{-nx}$ on $[0,infty)$

$x^n$ on $[0,1]$

$x/1+nx$ on $(0,1)$

$arctan(nx)$ on $mathbb{R}$

3 / 5

Is arc length preserved under uniform convergence

True

False

4 / 5

Is Area preserved under uniform convergence?

True

False

5 / 5

What is the pointwise limit for the sequence of functions $\arctan(nx)$ for $x > 0 $

0

$pi/2$

$pi/4$

$pi$

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